Proving the identity $sum_{k=1}^n {k^3} = big(sum_{k=1}^n kbig)^2$ without induction












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I recently proved that



$$sum_{k=1}^n k^3 = left(sum_{k=1}^n k right)^2$$



using mathematical induction. I'm interested if there's an intuitive explanation, or even a combinatorial interpretation of this property. I would also like to see any other proofs.










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  • 6




    $begingroup$
    math.stackexchange.com/questions/61798/…
    $endgroup$
    – Arjang
    Jun 27 '12 at 9:22










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    Look at this takayaiwamoto.com/Sums_and_Series/sumcube_1.html
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    – pritam
    Jun 27 '12 at 9:29










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    See math.stackexchange.com/questions/120674 for remarks about proofs "not using induction".
    $endgroup$
    – sdcvvc
    Jun 27 '12 at 10:13










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    I merged the three existing posts which covered exactly this question, as each post had different interesting answers which should not be lost. I also deleted redundant comments, and comments about closing posts as duplicates. This fourth question is not considered a duplicate.
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    – Eric Naslund
    Jul 2 '12 at 11:30








  • 2




    $begingroup$
    Since this question is asked frequently, it has been added to the list of Generalizations of Common questions. It has been kept seperate from the version which does use induction.
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    – Eric Naslund
    Aug 30 '12 at 0:23


















149












$begingroup$


I recently proved that



$$sum_{k=1}^n k^3 = left(sum_{k=1}^n k right)^2$$



using mathematical induction. I'm interested if there's an intuitive explanation, or even a combinatorial interpretation of this property. I would also like to see any other proofs.










share|cite|improve this question











$endgroup$








  • 6




    $begingroup$
    math.stackexchange.com/questions/61798/…
    $endgroup$
    – Arjang
    Jun 27 '12 at 9:22










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    Look at this takayaiwamoto.com/Sums_and_Series/sumcube_1.html
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    – pritam
    Jun 27 '12 at 9:29










  • $begingroup$
    See math.stackexchange.com/questions/120674 for remarks about proofs "not using induction".
    $endgroup$
    – sdcvvc
    Jun 27 '12 at 10:13










  • $begingroup$
    I merged the three existing posts which covered exactly this question, as each post had different interesting answers which should not be lost. I also deleted redundant comments, and comments about closing posts as duplicates. This fourth question is not considered a duplicate.
    $endgroup$
    – Eric Naslund
    Jul 2 '12 at 11:30








  • 2




    $begingroup$
    Since this question is asked frequently, it has been added to the list of Generalizations of Common questions. It has been kept seperate from the version which does use induction.
    $endgroup$
    – Eric Naslund
    Aug 30 '12 at 0:23
















149












149








149


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$begingroup$


I recently proved that



$$sum_{k=1}^n k^3 = left(sum_{k=1}^n k right)^2$$



using mathematical induction. I'm interested if there's an intuitive explanation, or even a combinatorial interpretation of this property. I would also like to see any other proofs.










share|cite|improve this question











$endgroup$




I recently proved that



$$sum_{k=1}^n k^3 = left(sum_{k=1}^n k right)^2$$



using mathematical induction. I'm interested if there's an intuitive explanation, or even a combinatorial interpretation of this property. I would also like to see any other proofs.







sequences-and-series algebra-precalculus summation visualization






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edited Oct 20 '17 at 22:37









Jack

27.5k1782203




27.5k1782203










asked Sep 2 '11 at 21:06









F MF M

3,09152341




3,09152341








  • 6




    $begingroup$
    math.stackexchange.com/questions/61798/…
    $endgroup$
    – Arjang
    Jun 27 '12 at 9:22










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    Look at this takayaiwamoto.com/Sums_and_Series/sumcube_1.html
    $endgroup$
    – pritam
    Jun 27 '12 at 9:29










  • $begingroup$
    See math.stackexchange.com/questions/120674 for remarks about proofs "not using induction".
    $endgroup$
    – sdcvvc
    Jun 27 '12 at 10:13










  • $begingroup$
    I merged the three existing posts which covered exactly this question, as each post had different interesting answers which should not be lost. I also deleted redundant comments, and comments about closing posts as duplicates. This fourth question is not considered a duplicate.
    $endgroup$
    – Eric Naslund
    Jul 2 '12 at 11:30








  • 2




    $begingroup$
    Since this question is asked frequently, it has been added to the list of Generalizations of Common questions. It has been kept seperate from the version which does use induction.
    $endgroup$
    – Eric Naslund
    Aug 30 '12 at 0:23
















  • 6




    $begingroup$
    math.stackexchange.com/questions/61798/…
    $endgroup$
    – Arjang
    Jun 27 '12 at 9:22










  • $begingroup$
    Look at this takayaiwamoto.com/Sums_and_Series/sumcube_1.html
    $endgroup$
    – pritam
    Jun 27 '12 at 9:29










  • $begingroup$
    See math.stackexchange.com/questions/120674 for remarks about proofs "not using induction".
    $endgroup$
    – sdcvvc
    Jun 27 '12 at 10:13










  • $begingroup$
    I merged the three existing posts which covered exactly this question, as each post had different interesting answers which should not be lost. I also deleted redundant comments, and comments about closing posts as duplicates. This fourth question is not considered a duplicate.
    $endgroup$
    – Eric Naslund
    Jul 2 '12 at 11:30








  • 2




    $begingroup$
    Since this question is asked frequently, it has been added to the list of Generalizations of Common questions. It has been kept seperate from the version which does use induction.
    $endgroup$
    – Eric Naslund
    Aug 30 '12 at 0:23










6




6




$begingroup$
math.stackexchange.com/questions/61798/…
$endgroup$
– Arjang
Jun 27 '12 at 9:22




$begingroup$
math.stackexchange.com/questions/61798/…
$endgroup$
– Arjang
Jun 27 '12 at 9:22












$begingroup$
Look at this takayaiwamoto.com/Sums_and_Series/sumcube_1.html
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– pritam
Jun 27 '12 at 9:29




$begingroup$
Look at this takayaiwamoto.com/Sums_and_Series/sumcube_1.html
$endgroup$
– pritam
Jun 27 '12 at 9:29












$begingroup$
See math.stackexchange.com/questions/120674 for remarks about proofs "not using induction".
$endgroup$
– sdcvvc
Jun 27 '12 at 10:13




$begingroup$
See math.stackexchange.com/questions/120674 for remarks about proofs "not using induction".
$endgroup$
– sdcvvc
Jun 27 '12 at 10:13












$begingroup$
I merged the three existing posts which covered exactly this question, as each post had different interesting answers which should not be lost. I also deleted redundant comments, and comments about closing posts as duplicates. This fourth question is not considered a duplicate.
$endgroup$
– Eric Naslund
Jul 2 '12 at 11:30






$begingroup$
I merged the three existing posts which covered exactly this question, as each post had different interesting answers which should not be lost. I also deleted redundant comments, and comments about closing posts as duplicates. This fourth question is not considered a duplicate.
$endgroup$
– Eric Naslund
Jul 2 '12 at 11:30






2




2




$begingroup$
Since this question is asked frequently, it has been added to the list of Generalizations of Common questions. It has been kept seperate from the version which does use induction.
$endgroup$
– Eric Naslund
Aug 30 '12 at 0:23






$begingroup$
Since this question is asked frequently, it has been added to the list of Generalizations of Common questions. It has been kept seperate from the version which does use induction.
$endgroup$
– Eric Naslund
Aug 30 '12 at 0:23












23 Answers
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Stare at the following image, taken from this MO answer, long enough:



Proof that the sum of the cubes is the square of the sum






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    original link: users.tru.eastlink.ca/~brsears/math/oldprob.htm#s32
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    – Foo Bah
    Sep 3 '11 at 3:36






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    The fact that there are $k$ blocks (or $frac{1}{2}+k{-}1+frac{1}{2}$ blocks) of $ktimes k$ size is based on the fact that $sumlimits_{j=1}^{k-1}=k(k{-}1)/2$. That is, $(k{-}1)/2$ blocks on top $(k{-}1)/2$ on the left and $1$ block at the corner (totaling to $k$). Perhaps I am being picky or slow, but I don't see this as obvious from the image. Beyond that, it is a nice proof-without-words.
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    – robjohn
    Sep 3 '11 at 4:13








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    I don’t think this is a proof free from induction.
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    – k.stm
    Mar 27 '14 at 12:58






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    This only proves the assertion for $n=5$. If someone is to accept its validity for general $n$, then it is clearly not induction-free.
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    – uniquesolution
    Oct 11 '15 at 0:39






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    That only shows you haven't stared at the image long enough...
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    – Mariano Suárez-Álvarez
    Oct 11 '15 at 2:23



















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I don't know if this is intuitive, but it is graphic.



Graphic proof that the sum of cubes is the square of the sum of first powers



On the outer edge of each $(k{+}1){times}k$ block there are $k$ pairs of products each of which total to $k^2$. Thus, the outer edge sums to $k^3$, and the sum of the whole array is therefore $sumlimits_{k=1}^n k^3$.



The array is the matrix product
$$
left[begin{array}{r}0\1\2\vdots\nend{array}right]bulletleft[begin{array}{rrrrr}1&2&3&cdots&nend{array}right]
$$
Therefore, the sum of the elements of the array is $sumlimits_{k=0}^nk;sumlimits_{k=1}^nk=left(sumlimits_{k=1}^nkright)^2$.



Therefore, $sumlimits_{k=1}^n k^3=left(sumlimits_{k=1}^nkright)^2$






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    If we forget the first line of the matrix (which is zero and which is only used to make pairs with the diagonal coefficients), then I like the fact we can put this answer in parallel with the coloured rectangles above and below, and we get another partition of each colored area (each coefficient of the matrix gives a rectangle of a certain area), which explains why each L-shaped area is $k^3$.
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    – Wok
    Sep 4 '11 at 7:39










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    However, each L-shape coefficient has the same factor $k$, which means it proves "each L-shaped area is $k^3$" by the same proof that $2 times sum_{j=0}^k j = (0+k) + (1+k-1) + ... + (k-1+1) + (k+0) = (k+1) times k$, which makes it really close to the coloured rectangles above and below.
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    – Wok
    Sep 4 '11 at 7:49










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    I hope you don't mind if I use both ideas in another answer.
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    – Wok
    Sep 4 '11 at 8:13










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    I don't mind. I simply find it less aesthetic to need to use $sumlimits_{j=1}^k;j=k(k+1)/2$ or that $(k(k+1)/2)^2-(k(k-1)/2)^2=k^3$ in an intuitive proof.
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    – robjohn
    Sep 4 '11 at 19:31





















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Can you get the intuition explanation from the following two pictures?[EDIT: the following is essentially the same as Mariano's answer. He didn't mentioned the first picture though.]



enter image description hereenter image description here



The images are from Brian R Sears.






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    nice, but already posted (linked) by Mariano
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    – leonbloy
    Sep 2 '11 at 23:24



















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I believe this illustration is due to Anders Kaseorg:



Visual proof by block-stacking






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    There's this nice picture from the Wikipedia entry on the squared triangular number:



    enter image description here



    The left side shows that $1 + 2 + 3$ forms a triangle and so that squaring it produces a larger triangle made up of $1+2+3$ copies of the original triangle. The right side has $1(1^2) + 2(2^2) + 3(3^2) = 1^3 + 2^3 + 3^3$. The coloring shows why the two sides are equal.



    There are several other references for combinatorial proofs and geometric arguments on the Wikipedia page.






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      Here's another version of this "proof without words". This is the case $n=4$.



      enter image description here



      There are 1 $1 times 1$, 2 $2 times 2$, 3 $3 times 3$, ... squares, for a total area of $1^3 + 2^3 + ldots + n^3$. For even $k$, two of the $k times k$ squares overlap in a $k/2 times k/2$ square, but this
      just balances out a $k/2 times k/2$ square that is left out, so the total is the area of
      a square of side $1 + 2 + ldots + n$.






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        Each colored area is $k^3$ as a difference of two areas: $S_k^2 - S_{k-1}^2$.



        enter image description here



        enter image description here





        The detailed proof which comes with the drawing is the following.



        For any positive integer $k$, we define:
        $$S_i = sum_{j=1}^{i} j$$



        We first notice:
        $$S_i^2 = S_i^2 - S_0^2= sum_{k=1}^{i} left(S_k^2 - S_{k-1}^2right)$$



        The expected result finally comes from:
        $$S_k^2 - S_{k-1}^2 = k left(k+2 S_{k-1}right) = kleft(k+kleft(k-1right)right)=k^3$$






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        • 1




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          So essentially, you are using the fact that $$left(sum_{j=1}^k;jright)^2-left(sum_{j=1}^{k-1};jright)^2=k^3$$ to justify the diagram which is supposed to prove that fact intuitively.
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          – robjohn
          Sep 4 '11 at 1:09












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          As you mentioned in another answer, this is where the diagram is the least intuitive. However, if you cannot picture $k^3$ on a 2D-plane, then you need another representation as a difference of areas.
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          – Wok
          Sep 4 '11 at 6:50










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          As soon as you know $S_k = sum_{j=1}^k j = frac{k(k+1)}{2}$, I find it more intuitive to figure out each colored area is $k^3$ on this diagram than to figure it out by counting squares plus two rectangles when $k$ is even.
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          – Wok
          Sep 4 '11 at 7:06





















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        The formula is due to Nicomachus of Gerasa. There is a nice discussion of ways to prove it at this n-category cafe post, including a bijective proof and some visual / "geometric" proofs.






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          Several visual proofs of this indentity are collected in the book
          Roger B. Nelsen: Proofs without Words
          starting from p.84.



          Although several of these proofs can still be considered inductive, I thought it might be interesting to mention them.



          Original sources are given on p. 147:




          • 84 Mathematical Gazette, vol. 49, no. 368 (May 1965), p. 199. jstor

          • 85 Mathematics Magazine, vol. 50, no. 2 (March 1977), p. 74. jstor

          • 86 Mathematics Magazine, vol. 58, no. 1 (Jan. 1985), p. 11. jstor

          • 87 Mathematics Magazine, vol. 62, no. 4 (Oct. 1989), p. 259. jstor

          • 87 Mathematical Gazette, vol. 49, no. 368 (May 1965), p. 200. jstor

          • 88 Mathematics Magazine, vol. 63, no. 3 (June 1990), p. 178. jstor

          • 89 Mathematics Magazine, vol. 62, no. 5 (Dec. 1989), p. 323. jstor

          • 90 Mathematics Magazine, vol. 65, no. 3 (June 1992), p. 185. jstor






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            Great collections. Thanks
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            – mrs
            Jun 27 '12 at 11:11



















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          The sum of a degree $n$ polynomial $f(n)$ will be a degree $n+1$ polynomial $S(n)$ for $n geq 0$ and both polynomials can be extended (maintaining the relation $S(n)-S(n-1) = f(n)$) to negative $n$.
          To verify that the formula for $Sigma k^3$ is correct one need only test it for any 5 distinct values of $n$, but the structure of the answer can be predicted algebraically using the continuation to negative $n$.



          If $S(n) = (1^3 + 2^3 + dots n^3)$ is the polynomial that satisfies $S(n)-S(n-1) = n^3$ and $S(1)=1$, then one can calculate from that equation that $S(0)=S(-1)=0$ and $S(-n-1)=S(n)$ for all negative $n$, so that $S$ is symmetric around $-1/2$. The vanishing at 0 and -1 implies that $S(t)$ is divisible as a polynomial by $t(t+1)$. The symmetry implies that $S(t)$ is a function (necessarily a polynomial) of $t(t+1)$.



          $S(t)$ being of degree 4, this means $S(n) = a (n)(n+1) + b((n^2 +n)^2$ for constants $a$ and $b$. Summation being analogous to integration (and equal to it in a suitable limit), they have to agree on highest degree terms. Here it forces $b$ to be $1/4$ to match $int x^3 = x^4/4$. Computing the sum at a single point such as $n=1$ determines $a$, which is zero.



          Similar reasoning shows that $S_k(n)$ is divisible as a polynomial by $n(n+1)$ for all $k$. For odd $k$, $S_k(n)$ is a polynomial in $n(n+1)$.






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            Here's a direct algebraic proof. $$sum_{k=1}^n(k^3-k^2)=2sum_{k=1}^nkcdotfrac{k(k-1)}2=2sum_{k=1}^nksum_{l=1}^{k-1}l=2sum_{1leqslant l<kleqslant n}kl=left(sum_{k=1}^nkright)^2-sum_{k=1}^nk^2$$






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              You know, $sum_0^n x^k=frac{1-x^{n+1}}{1-x}$. Differentiate both sides once, $sum_1^n kx^{k-1}=frac{x^n(nx-n-1)+1}{x^2-2x+1}$. Now taking $lim_{xto1}$ both sides and then squaring the result will give you the expression on the RHS. You can further differentiate $sum_0^n x^k=frac{1-x^{n+1}}{1-x}$ until you get $k^3$ inside the expression, take limit again you will get the same result as of $(lim_{xto1}frac{x^n(nx-n-1)+1}{x^2-2x+1})^2$. You can also prove it using telescopic series.






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                why does the assumption hold? this is usually proved using induction...
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                – akkkk
                Jun 27 '12 at 9:47










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                what assumption??
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                – Aang
                Jun 27 '12 at 9:48










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                I guess Auke means $sum_{0}^{n} x^k=frac{1-x^{n+1}}{1-x}$.
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                – sdcvvc
                Jun 27 '12 at 10:12












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                LHS is a geometric series. en.wikipedia.org/wiki/Geometric_progression
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                – Aang
                Jun 27 '12 at 10:14








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                @Auke: one can also prove it like this - let $f(x) = sumlimits_{k=0}^nx^k$, then $f(x) - xf(x)$ is: $$ begin{align} 1+&x+x^2+dots+x^n \ &- \ &x+x^2+dots+x^n+x^{n+1} end{align} $$ which is $1-x^{n+1}$. Hence $$ (1-x)f(x) = 1-x^{n+1} $$ as needed.
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                – Ilya
                Jun 27 '12 at 12:40





















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              We know that $$A=sum_1^n k^3=frac{1}{4}n^4+frac{1}{2}n^3+frac{1}{4}n^2$$ and $$B=sum_1^n k=frac{1}{2}n^2+frac{1}{2}n$$ $A-B^2=0$. :)






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                If you are presenting this proof,write it nicely. $A=frac{n^2(n+1)^2}{4}$ and $B=frac{n(n+1)}{2}$,obviously $A=B^2$
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                – Aang
                Jun 27 '12 at 10:40










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                @avatar: Dear Avatar; I will. Your proof is great and it looks analytically. Thanks. :-)
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                – mrs
                Jun 27 '12 at 10:48










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                sorry, if you felt that i was being rude.
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                – Aang
                Jun 27 '12 at 10:53










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                @avatar: Your proof lights mine. No Problem at all. WELCOME Avatar. :) :)
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                – mrs
                Jun 27 '12 at 10:57



















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              $f(n)=1^3+2^3+3^3+cdots+n^3$



              $f(n-1)=1^3+2^3+3^3+cdots+(n-1)^3$



              $f(n)-f(n-1)=n^3$



              if $f(n)= (1+2+3+4+cdots+n)^2$ then



              $$f(n)-f(n-1)=(1+2+3+4+cdots+n)^2-(1+2+3+4+cdots+(n-1))^2$$



              using $a^2-b^2=(a+b)(a-b)$



              $f(n)-f(n-1)=$



              $=[1+1+2+2+3+3+4+4+cdots+(n-1)+(n-1)+n][1-1+2-2+3-3+4-4+cdots+(n-1)-(n-1)+n]=$



              $=[2(1+2+3+4+cdots+(n-1))+n]n=(2frac{n(n-1)}{2}+n)n=(n(n-1)+n)n=n^3$






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                http://en.wikipedia.org/wiki/Faulhaber%27s_formula#Faulhaber_polynomials



                If $p$ is odd, then $1^p+2^p+3^p+cdots+n^p$ is a polynomial function of $a=1+2+3+cdots+n$. If $p=3$, then then the sum is $a^2$; if $p=5$ then it's $(4a^3-a^2)/3$, and so on.






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                  And $a$ is always a factor of $p$.
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                  – lhf
                  Sep 3 '11 at 13:26



















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                Chance would have it that I stumbled* upon this article today:



                http://blogs.mathworks.com/loren/2010/03/04/nichomachuss-theorem/



                It seems to answer your question.



                (* That is, @AlgebraFact on Twitter posted a link)






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                  The square in the identity is the area of the triangle below, while the cubes are the area's of the trapezoidal layers, with heights $k = 1, 2, cdots, n$
                  TriangleWithHeight 1+2+..+n



                  The trapezoids have area $k^3$ because their height equals $k$ and the $text{width}_{text{atHalfHeight}}$ consists of $k$ diagonals with width $k$:
                  Trapezoidal layer with height = k and width = k^2



                  The total of the triangle is its squared height $(1 + 2 + cdots + n)^2$, because this triangle can be turned into a square:
                  The triangle cut in two and recomposed as a square.



                  Therefore:
                  $(1 + 2 + cdots + n)^2 = sum_{k=1}^n k^3$ , $blacksquare$






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                    One way to show that
                    $$sum_{i=1}^n i^3 = bigg(sum_{i=1}^n i bigg)^2$$
                    is to add up the entries in the multiplication tables, but first we need to show that
                    $$1+2+3+dots+n+dots+3+2+1 = n^2$$
                    For this, see the image below (n=7)enter image description here
                    $$7^2=color{green}{1+2+3+4+5+6+7}color{red}{+6+5+4+3+2+1}$$
                    Next, consider the standard multiplication table that we are all familiar with.The graphic shows the table up to the 9s.



                    enter image description here



                    We can add up the entries in any order that we wish.
                    One way would be to add up a series of Ls (the 6th L ($L_6$) is highlighted in yellow).
                    $$begin{align}
                    L_6 &= 6+12+18+24+30+36+30+24+18+12+6\
                    &=6(1+2+3+4+5+6+5+4+3+2+1)\
                    &=6(6^2)\
                    &=6^3
                    end{align}$$
                    And the sum of all the entries in the table becomes
                    $$sum_{i=1}^n L_i = sum_{i=1}^n i^3$$
                    Alternatively, we could just add up each row. The 6th row (R_6) would be
                    $$begin{align}
                    R_6 &= 6+12+18+24+30+36+42+48+54\
                    &= 6(1+2+3+4+5+6+7+8+9)\
                    &= 6sum_{i=1}^9 i
                    end{align}$$
                    And the sum of all the entries becomes
                    $$sum_{i=1}^n R_i = sum_{i=1}^n bigg(isum_{j=1}^n jbigg)=bigg(sum_{j=1}^n jbigg)bigg(sum_{i=1}^n ibigg)=bigg(sum_{i=1}^n ibigg)^2$$
                    Thus we have
                    $$sum_{i=1}^n i^3 = sum_{i=1}^n L_i = sum_{i=1}^n R_i=bigg(sum_{i=1}^n ibigg)^2$$






                    share|cite|improve this answer











                    $endgroup$





















                      4












                      $begingroup$

                      square triangular proof without words



                      This is about the same proof as here, the presentation is a bit different though. This is another way to make $k^3$ appear than what was shown here, here and here.






                      share|cite|improve this answer











                      $endgroup$





















                        4












                        $begingroup$

                        Here's a simple bijective proof of a different sort:



                        Consider a staircase with $n$ steps, built out of $sum_{k=1}^n k$ blocks. In other words, take the set ${(i,j) in mathbb{Z}timesmathbb{Z}: i + j leq n, i > 0, j > 0}$.



                        Then $left(sum_{k=1}^n kright)^2$ is the number of ordered pairs $(B_1,B_2)$ of blocks.



                        And $sum_{k=1}^n k^3$ is the number of ordered $4$-tuples $(k,b_1,b_2,b_3)$, where $k in {1,ldots,n}$, and $b_1$ is along the $k$-th diagonal $b_1 in {(k+1-j,j): j in {1,ldots,k}}$, and $b_2$ is along the bottom $b_2 in {(j,1): j in {1,ldots, k}}$ and $b_3$ is along the left side $b_3 in {(1,j): j in {1, ldots, k}}$.



                        The bijection:



                        Given an ordered tuple $(k,b_1,b_2,b_3)$, let $A_1 = b_1$ and let $A_2$ given by $b_2$ and $b_3$ as its $x$ and $y$ coordinates, so if $b_2 = (i,1)$ and $b_3 = (1,j)$ then $A_2 = (i,j)$.



                        Case 1: $A_2$ is on or below the $k$-th diagonal. Then let $(B_1, B_2) = (A_1, A_2)$.



                        Case 2: $A_2$ is above the $k$-the diagonal. Then let $A_2'$ be the reflection across the $k$-th diagonal of $A_2$. That is, if $A_2 = (i,j)$ then $A_2' = (k+1-j,k+1-i)$. Then let $(B_1, B_2) = (A_2', A_1)$.



                        The inverse:



                        To get the inverse, take whichever of $B_1$ and $B_2$ is on a higher diagonal (i.e. has greater sum of its coordinates), taking $B_1$ in case of ties, and let that be $b_1$ and let $k$ be the sum of the coordinates of $b_1$.



                        Case 1: If $B_1$ is used: Take $B_2$ and let $b_2$ and $b_3$ be given by points with the same the $x$- and $y$-coordinates, respectively, as $B_2$.



                        Case 2: If $B_2$ is used: Take $B_1'$ (i.e. the reflection across the $k$-th diagonal, as above) and let $b_2$ and $b_3$ be given by points with the same the $x$- and $y$-coordinates, respectively, as $B_1'$.






                        share|cite|improve this answer











                        $endgroup$





















                          0












                          $begingroup$

                          After many years I still think the best way to solve this kind of problem in a natural and systematic way is to view it as a recurrence relation with constants coefficients, in this case, $x_n = x_{n-1}+n^3$. The way I learnt to do so was by using characteristic polynomial but you may prefer some other method...






                          share|cite|improve this answer











                          $endgroup$





















                            0












                            $begingroup$

                            For every $kinmathbb{N}$
                            $$(k+1)^4=k^4+4k^3++6k^2+4k+1$$
                            therefore
                            $$sum_{k=1}^n(k+1)^4=sum_{k=1}^nk^4+4sum_{k=1}^nk^3+6sum_{k=1}^nk^2+4sum_{k=1}^nk+sum_{k=1}^n1$$
                            which is equivalent to
                            $$sum_{k=1}^nk^4+(n+1)^4-1=sum_{k=1}^nk^4+4sum_{k=1}^nk^3+6sum_{k=1}^nk^2+2n(n+1)+n$$
                            After simplifications we obtain
                            $$4sum_{k=1}^nk^3=(n+1)^4-1-2n(n+1)-n-6sum_{k=1}^nk^2=n^4+4n^3+4n^2+n-6sum_{k=1}^nk^2$$
                            Using
                            $$sum_{k=1}^nk^2=frac{n(n+1)(2n+1)}{6}hspace{0.2cm}text{and}hspace{0.2cm}sum_{k=1}^nk=frac{n(n+1)}{2}$$
                            we get
                            $$4sum_{k=1}^nk^3=n^4+4n^3+4n^2+n-6sum_{k=1}^nk^2\=n^4+4n^3+4n^2+n-n(n+1)(2n+1)\=n^4+2n^3+n^2=n^2(n+1)^2$$
                            Finally
                            $$sum_{k=1}^nk^3=frac{n^2(n+1)^2}{4}=Big(frac{n(n+1)}{2}Big)^2=Big(sum_{k=1}^nkBig)^2$$






                            share|cite|improve this answer









                            $endgroup$





















                              0












                              $begingroup$

                              We begin by writing $k^3$ in a more clever fashion: $k^3 = k(k-1)(k-2) + 3k^2 - 2k$ :



                              $$sum_{k=0}^n k^3 = sum_{k=0}^n k(k-1)(k-2) + 3k^2 -2k$$



                              Distributing the summation we obtain: $$ sum_{k=0}^n k(k-1)(k-2) + sum_{k=0}^n 3k^2 - sum_{k=0}^n 2k$$



                              Notice, $$k(k-1)(k-2) = frac {k!}{(k-3)!}$$
                              Now we have
                              $$sum_{k=0}^n frac {k!}{(k-3)!} + sum_{k=0}^n 3k^2 - sum_{k=0}^n 2k$$
                              Notice that we have nearly obtained the binomial expansion of K choose 3, all we need to do is divide by 3! So we offset this by also taking the product of 3!
                              $$sum_{k=0}^n frac {k!}{(k-3)!} = 3!sum_{k=0}^n frac {k!}{(k-3)!3!} = 3!sum_{k=0}^nbinom{k}{3} = 3!binom{n+1}{4}$$
                              We have now obtained
                              $$sum_{k=0}^n k^3 = 3!binom{n+1}{4} + 3sum_{k=0}^n k^2 - 2sum_{k=0}^n k$$
                              Focusing solely on the right-hand side we have
                              $$6biggl(frac {(n+1)!}{(n-3)!4!}biggr) + 3sum_{k=0}^n k^2 - 2sum_{k=0}^n k$$
                              Assuming we already know the sum of the sequence of integers and squared integers (the 2 sums we have left) we have
                              $$ frac {(n+1)(n)(n-1)(n-2)}{4} + 3frac{n(n+1)(2n+1)}{6} - 2frac {n(n+1)}{2}$$
                              Generating common denominators and with a bit of algebra we now have
                              $$ frac {n^4-2n^3-n^2+2n+4n^3+6n^2+2n-4n^2-4n}{4}$$
                              Combining like-terms we have reached our solution:
                              $$ frac {n^4+2n^3+n^2}{4} = biggl(sum_{k=0}^n kbiggr)^2$$






                              share|cite|improve this answer











                              $endgroup$













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                                143












                                $begingroup$

                                Stare at the following image, taken from this MO answer, long enough:



                                Proof that the sum of the cubes is the square of the sum






                                share|cite|improve this answer











                                $endgroup$









                                • 4




                                  $begingroup$
                                  original link: users.tru.eastlink.ca/~brsears/math/oldprob.htm#s32
                                  $endgroup$
                                  – Foo Bah
                                  Sep 3 '11 at 3:36






                                • 15




                                  $begingroup$
                                  The fact that there are $k$ blocks (or $frac{1}{2}+k{-}1+frac{1}{2}$ blocks) of $ktimes k$ size is based on the fact that $sumlimits_{j=1}^{k-1}=k(k{-}1)/2$. That is, $(k{-}1)/2$ blocks on top $(k{-}1)/2$ on the left and $1$ block at the corner (totaling to $k$). Perhaps I am being picky or slow, but I don't see this as obvious from the image. Beyond that, it is a nice proof-without-words.
                                  $endgroup$
                                  – robjohn
                                  Sep 3 '11 at 4:13








                                • 13




                                  $begingroup$
                                  I don’t think this is a proof free from induction.
                                  $endgroup$
                                  – k.stm
                                  Mar 27 '14 at 12:58






                                • 3




                                  $begingroup$
                                  This only proves the assertion for $n=5$. If someone is to accept its validity for general $n$, then it is clearly not induction-free.
                                  $endgroup$
                                  – uniquesolution
                                  Oct 11 '15 at 0:39






                                • 4




                                  $begingroup$
                                  That only shows you haven't stared at the image long enough...
                                  $endgroup$
                                  – Mariano Suárez-Álvarez
                                  Oct 11 '15 at 2:23
















                                143












                                $begingroup$

                                Stare at the following image, taken from this MO answer, long enough:



                                Proof that the sum of the cubes is the square of the sum






                                share|cite|improve this answer











                                $endgroup$









                                • 4




                                  $begingroup$
                                  original link: users.tru.eastlink.ca/~brsears/math/oldprob.htm#s32
                                  $endgroup$
                                  – Foo Bah
                                  Sep 3 '11 at 3:36






                                • 15




                                  $begingroup$
                                  The fact that there are $k$ blocks (or $frac{1}{2}+k{-}1+frac{1}{2}$ blocks) of $ktimes k$ size is based on the fact that $sumlimits_{j=1}^{k-1}=k(k{-}1)/2$. That is, $(k{-}1)/2$ blocks on top $(k{-}1)/2$ on the left and $1$ block at the corner (totaling to $k$). Perhaps I am being picky or slow, but I don't see this as obvious from the image. Beyond that, it is a nice proof-without-words.
                                  $endgroup$
                                  – robjohn
                                  Sep 3 '11 at 4:13








                                • 13




                                  $begingroup$
                                  I don’t think this is a proof free from induction.
                                  $endgroup$
                                  – k.stm
                                  Mar 27 '14 at 12:58






                                • 3




                                  $begingroup$
                                  This only proves the assertion for $n=5$. If someone is to accept its validity for general $n$, then it is clearly not induction-free.
                                  $endgroup$
                                  – uniquesolution
                                  Oct 11 '15 at 0:39






                                • 4




                                  $begingroup$
                                  That only shows you haven't stared at the image long enough...
                                  $endgroup$
                                  – Mariano Suárez-Álvarez
                                  Oct 11 '15 at 2:23














                                143












                                143








                                143





                                $begingroup$

                                Stare at the following image, taken from this MO answer, long enough:



                                Proof that the sum of the cubes is the square of the sum






                                share|cite|improve this answer











                                $endgroup$



                                Stare at the following image, taken from this MO answer, long enough:



                                Proof that the sum of the cubes is the square of the sum







                                share|cite|improve this answer














                                share|cite|improve this answer



                                share|cite|improve this answer








                                edited Sep 20 '17 at 3:43









                                Parcly Taxel

                                44.7k1376109




                                44.7k1376109










                                answered Sep 2 '11 at 21:16









                                Mariano Suárez-ÁlvarezMariano Suárez-Álvarez

                                112k7157289




                                112k7157289








                                • 4




                                  $begingroup$
                                  original link: users.tru.eastlink.ca/~brsears/math/oldprob.htm#s32
                                  $endgroup$
                                  – Foo Bah
                                  Sep 3 '11 at 3:36






                                • 15




                                  $begingroup$
                                  The fact that there are $k$ blocks (or $frac{1}{2}+k{-}1+frac{1}{2}$ blocks) of $ktimes k$ size is based on the fact that $sumlimits_{j=1}^{k-1}=k(k{-}1)/2$. That is, $(k{-}1)/2$ blocks on top $(k{-}1)/2$ on the left and $1$ block at the corner (totaling to $k$). Perhaps I am being picky or slow, but I don't see this as obvious from the image. Beyond that, it is a nice proof-without-words.
                                  $endgroup$
                                  – robjohn
                                  Sep 3 '11 at 4:13








                                • 13




                                  $begingroup$
                                  I don’t think this is a proof free from induction.
                                  $endgroup$
                                  – k.stm
                                  Mar 27 '14 at 12:58






                                • 3




                                  $begingroup$
                                  This only proves the assertion for $n=5$. If someone is to accept its validity for general $n$, then it is clearly not induction-free.
                                  $endgroup$
                                  – uniquesolution
                                  Oct 11 '15 at 0:39






                                • 4




                                  $begingroup$
                                  That only shows you haven't stared at the image long enough...
                                  $endgroup$
                                  – Mariano Suárez-Álvarez
                                  Oct 11 '15 at 2:23














                                • 4




                                  $begingroup$
                                  original link: users.tru.eastlink.ca/~brsears/math/oldprob.htm#s32
                                  $endgroup$
                                  – Foo Bah
                                  Sep 3 '11 at 3:36






                                • 15




                                  $begingroup$
                                  The fact that there are $k$ blocks (or $frac{1}{2}+k{-}1+frac{1}{2}$ blocks) of $ktimes k$ size is based on the fact that $sumlimits_{j=1}^{k-1}=k(k{-}1)/2$. That is, $(k{-}1)/2$ blocks on top $(k{-}1)/2$ on the left and $1$ block at the corner (totaling to $k$). Perhaps I am being picky or slow, but I don't see this as obvious from the image. Beyond that, it is a nice proof-without-words.
                                  $endgroup$
                                  – robjohn
                                  Sep 3 '11 at 4:13








                                • 13




                                  $begingroup$
                                  I don’t think this is a proof free from induction.
                                  $endgroup$
                                  – k.stm
                                  Mar 27 '14 at 12:58






                                • 3




                                  $begingroup$
                                  This only proves the assertion for $n=5$. If someone is to accept its validity for general $n$, then it is clearly not induction-free.
                                  $endgroup$
                                  – uniquesolution
                                  Oct 11 '15 at 0:39






                                • 4




                                  $begingroup$
                                  That only shows you haven't stared at the image long enough...
                                  $endgroup$
                                  – Mariano Suárez-Álvarez
                                  Oct 11 '15 at 2:23








                                4




                                4




                                $begingroup$
                                original link: users.tru.eastlink.ca/~brsears/math/oldprob.htm#s32
                                $endgroup$
                                – Foo Bah
                                Sep 3 '11 at 3:36




                                $begingroup$
                                original link: users.tru.eastlink.ca/~brsears/math/oldprob.htm#s32
                                $endgroup$
                                – Foo Bah
                                Sep 3 '11 at 3:36




                                15




                                15




                                $begingroup$
                                The fact that there are $k$ blocks (or $frac{1}{2}+k{-}1+frac{1}{2}$ blocks) of $ktimes k$ size is based on the fact that $sumlimits_{j=1}^{k-1}=k(k{-}1)/2$. That is, $(k{-}1)/2$ blocks on top $(k{-}1)/2$ on the left and $1$ block at the corner (totaling to $k$). Perhaps I am being picky or slow, but I don't see this as obvious from the image. Beyond that, it is a nice proof-without-words.
                                $endgroup$
                                – robjohn
                                Sep 3 '11 at 4:13






                                $begingroup$
                                The fact that there are $k$ blocks (or $frac{1}{2}+k{-}1+frac{1}{2}$ blocks) of $ktimes k$ size is based on the fact that $sumlimits_{j=1}^{k-1}=k(k{-}1)/2$. That is, $(k{-}1)/2$ blocks on top $(k{-}1)/2$ on the left and $1$ block at the corner (totaling to $k$). Perhaps I am being picky or slow, but I don't see this as obvious from the image. Beyond that, it is a nice proof-without-words.
                                $endgroup$
                                – robjohn
                                Sep 3 '11 at 4:13






                                13




                                13




                                $begingroup$
                                I don’t think this is a proof free from induction.
                                $endgroup$
                                – k.stm
                                Mar 27 '14 at 12:58




                                $begingroup$
                                I don’t think this is a proof free from induction.
                                $endgroup$
                                – k.stm
                                Mar 27 '14 at 12:58




                                3




                                3




                                $begingroup$
                                This only proves the assertion for $n=5$. If someone is to accept its validity for general $n$, then it is clearly not induction-free.
                                $endgroup$
                                – uniquesolution
                                Oct 11 '15 at 0:39




                                $begingroup$
                                This only proves the assertion for $n=5$. If someone is to accept its validity for general $n$, then it is clearly not induction-free.
                                $endgroup$
                                – uniquesolution
                                Oct 11 '15 at 0:39




                                4




                                4




                                $begingroup$
                                That only shows you haven't stared at the image long enough...
                                $endgroup$
                                – Mariano Suárez-Álvarez
                                Oct 11 '15 at 2:23




                                $begingroup$
                                That only shows you haven't stared at the image long enough...
                                $endgroup$
                                – Mariano Suárez-Álvarez
                                Oct 11 '15 at 2:23











                                40












                                $begingroup$

                                I don't know if this is intuitive, but it is graphic.



                                Graphic proof that the sum of cubes is the square of the sum of first powers



                                On the outer edge of each $(k{+}1){times}k$ block there are $k$ pairs of products each of which total to $k^2$. Thus, the outer edge sums to $k^3$, and the sum of the whole array is therefore $sumlimits_{k=1}^n k^3$.



                                The array is the matrix product
                                $$
                                left[begin{array}{r}0\1\2\vdots\nend{array}right]bulletleft[begin{array}{rrrrr}1&2&3&cdots&nend{array}right]
                                $$
                                Therefore, the sum of the elements of the array is $sumlimits_{k=0}^nk;sumlimits_{k=1}^nk=left(sumlimits_{k=1}^nkright)^2$.



                                Therefore, $sumlimits_{k=1}^n k^3=left(sumlimits_{k=1}^nkright)^2$






                                share|cite|improve this answer











                                $endgroup$













                                • $begingroup$
                                  If we forget the first line of the matrix (which is zero and which is only used to make pairs with the diagonal coefficients), then I like the fact we can put this answer in parallel with the coloured rectangles above and below, and we get another partition of each colored area (each coefficient of the matrix gives a rectangle of a certain area), which explains why each L-shaped area is $k^3$.
                                  $endgroup$
                                  – Wok
                                  Sep 4 '11 at 7:39










                                • $begingroup$
                                  However, each L-shape coefficient has the same factor $k$, which means it proves "each L-shaped area is $k^3$" by the same proof that $2 times sum_{j=0}^k j = (0+k) + (1+k-1) + ... + (k-1+1) + (k+0) = (k+1) times k$, which makes it really close to the coloured rectangles above and below.
                                  $endgroup$
                                  – Wok
                                  Sep 4 '11 at 7:49










                                • $begingroup$
                                  I hope you don't mind if I use both ideas in another answer.
                                  $endgroup$
                                  – Wok
                                  Sep 4 '11 at 8:13










                                • $begingroup$
                                  I don't mind. I simply find it less aesthetic to need to use $sumlimits_{j=1}^k;j=k(k+1)/2$ or that $(k(k+1)/2)^2-(k(k-1)/2)^2=k^3$ in an intuitive proof.
                                  $endgroup$
                                  – robjohn
                                  Sep 4 '11 at 19:31


















                                40












                                $begingroup$

                                I don't know if this is intuitive, but it is graphic.



                                Graphic proof that the sum of cubes is the square of the sum of first powers



                                On the outer edge of each $(k{+}1){times}k$ block there are $k$ pairs of products each of which total to $k^2$. Thus, the outer edge sums to $k^3$, and the sum of the whole array is therefore $sumlimits_{k=1}^n k^3$.



                                The array is the matrix product
                                $$
                                left[begin{array}{r}0\1\2\vdots\nend{array}right]bulletleft[begin{array}{rrrrr}1&2&3&cdots&nend{array}right]
                                $$
                                Therefore, the sum of the elements of the array is $sumlimits_{k=0}^nk;sumlimits_{k=1}^nk=left(sumlimits_{k=1}^nkright)^2$.



                                Therefore, $sumlimits_{k=1}^n k^3=left(sumlimits_{k=1}^nkright)^2$






                                share|cite|improve this answer











                                $endgroup$













                                • $begingroup$
                                  If we forget the first line of the matrix (which is zero and which is only used to make pairs with the diagonal coefficients), then I like the fact we can put this answer in parallel with the coloured rectangles above and below, and we get another partition of each colored area (each coefficient of the matrix gives a rectangle of a certain area), which explains why each L-shaped area is $k^3$.
                                  $endgroup$
                                  – Wok
                                  Sep 4 '11 at 7:39










                                • $begingroup$
                                  However, each L-shape coefficient has the same factor $k$, which means it proves "each L-shaped area is $k^3$" by the same proof that $2 times sum_{j=0}^k j = (0+k) + (1+k-1) + ... + (k-1+1) + (k+0) = (k+1) times k$, which makes it really close to the coloured rectangles above and below.
                                  $endgroup$
                                  – Wok
                                  Sep 4 '11 at 7:49










                                • $begingroup$
                                  I hope you don't mind if I use both ideas in another answer.
                                  $endgroup$
                                  – Wok
                                  Sep 4 '11 at 8:13










                                • $begingroup$
                                  I don't mind. I simply find it less aesthetic to need to use $sumlimits_{j=1}^k;j=k(k+1)/2$ or that $(k(k+1)/2)^2-(k(k-1)/2)^2=k^3$ in an intuitive proof.
                                  $endgroup$
                                  – robjohn
                                  Sep 4 '11 at 19:31
















                                40












                                40








                                40





                                $begingroup$

                                I don't know if this is intuitive, but it is graphic.



                                Graphic proof that the sum of cubes is the square of the sum of first powers



                                On the outer edge of each $(k{+}1){times}k$ block there are $k$ pairs of products each of which total to $k^2$. Thus, the outer edge sums to $k^3$, and the sum of the whole array is therefore $sumlimits_{k=1}^n k^3$.



                                The array is the matrix product
                                $$
                                left[begin{array}{r}0\1\2\vdots\nend{array}right]bulletleft[begin{array}{rrrrr}1&2&3&cdots&nend{array}right]
                                $$
                                Therefore, the sum of the elements of the array is $sumlimits_{k=0}^nk;sumlimits_{k=1}^nk=left(sumlimits_{k=1}^nkright)^2$.



                                Therefore, $sumlimits_{k=1}^n k^3=left(sumlimits_{k=1}^nkright)^2$






                                share|cite|improve this answer











                                $endgroup$



                                I don't know if this is intuitive, but it is graphic.



                                Graphic proof that the sum of cubes is the square of the sum of first powers



                                On the outer edge of each $(k{+}1){times}k$ block there are $k$ pairs of products each of which total to $k^2$. Thus, the outer edge sums to $k^3$, and the sum of the whole array is therefore $sumlimits_{k=1}^n k^3$.



                                The array is the matrix product
                                $$
                                left[begin{array}{r}0\1\2\vdots\nend{array}right]bulletleft[begin{array}{rrrrr}1&2&3&cdots&nend{array}right]
                                $$
                                Therefore, the sum of the elements of the array is $sumlimits_{k=0}^nk;sumlimits_{k=1}^nk=left(sumlimits_{k=1}^nkright)^2$.



                                Therefore, $sumlimits_{k=1}^n k^3=left(sumlimits_{k=1}^nkright)^2$







                                share|cite|improve this answer














                                share|cite|improve this answer



                                share|cite|improve this answer








                                edited Sep 2 '11 at 22:30

























                                answered Sep 2 '11 at 21:55









                                robjohnrobjohn

                                269k27311638




                                269k27311638












                                • $begingroup$
                                  If we forget the first line of the matrix (which is zero and which is only used to make pairs with the diagonal coefficients), then I like the fact we can put this answer in parallel with the coloured rectangles above and below, and we get another partition of each colored area (each coefficient of the matrix gives a rectangle of a certain area), which explains why each L-shaped area is $k^3$.
                                  $endgroup$
                                  – Wok
                                  Sep 4 '11 at 7:39










                                • $begingroup$
                                  However, each L-shape coefficient has the same factor $k$, which means it proves "each L-shaped area is $k^3$" by the same proof that $2 times sum_{j=0}^k j = (0+k) + (1+k-1) + ... + (k-1+1) + (k+0) = (k+1) times k$, which makes it really close to the coloured rectangles above and below.
                                  $endgroup$
                                  – Wok
                                  Sep 4 '11 at 7:49










                                • $begingroup$
                                  I hope you don't mind if I use both ideas in another answer.
                                  $endgroup$
                                  – Wok
                                  Sep 4 '11 at 8:13










                                • $begingroup$
                                  I don't mind. I simply find it less aesthetic to need to use $sumlimits_{j=1}^k;j=k(k+1)/2$ or that $(k(k+1)/2)^2-(k(k-1)/2)^2=k^3$ in an intuitive proof.
                                  $endgroup$
                                  – robjohn
                                  Sep 4 '11 at 19:31




















                                • $begingroup$
                                  If we forget the first line of the matrix (which is zero and which is only used to make pairs with the diagonal coefficients), then I like the fact we can put this answer in parallel with the coloured rectangles above and below, and we get another partition of each colored area (each coefficient of the matrix gives a rectangle of a certain area), which explains why each L-shaped area is $k^3$.
                                  $endgroup$
                                  – Wok
                                  Sep 4 '11 at 7:39










                                • $begingroup$
                                  However, each L-shape coefficient has the same factor $k$, which means it proves "each L-shaped area is $k^3$" by the same proof that $2 times sum_{j=0}^k j = (0+k) + (1+k-1) + ... + (k-1+1) + (k+0) = (k+1) times k$, which makes it really close to the coloured rectangles above and below.
                                  $endgroup$
                                  – Wok
                                  Sep 4 '11 at 7:49










                                • $begingroup$
                                  I hope you don't mind if I use both ideas in another answer.
                                  $endgroup$
                                  – Wok
                                  Sep 4 '11 at 8:13










                                • $begingroup$
                                  I don't mind. I simply find it less aesthetic to need to use $sumlimits_{j=1}^k;j=k(k+1)/2$ or that $(k(k+1)/2)^2-(k(k-1)/2)^2=k^3$ in an intuitive proof.
                                  $endgroup$
                                  – robjohn
                                  Sep 4 '11 at 19:31


















                                $begingroup$
                                If we forget the first line of the matrix (which is zero and which is only used to make pairs with the diagonal coefficients), then I like the fact we can put this answer in parallel with the coloured rectangles above and below, and we get another partition of each colored area (each coefficient of the matrix gives a rectangle of a certain area), which explains why each L-shaped area is $k^3$.
                                $endgroup$
                                – Wok
                                Sep 4 '11 at 7:39




                                $begingroup$
                                If we forget the first line of the matrix (which is zero and which is only used to make pairs with the diagonal coefficients), then I like the fact we can put this answer in parallel with the coloured rectangles above and below, and we get another partition of each colored area (each coefficient of the matrix gives a rectangle of a certain area), which explains why each L-shaped area is $k^3$.
                                $endgroup$
                                – Wok
                                Sep 4 '11 at 7:39












                                $begingroup$
                                However, each L-shape coefficient has the same factor $k$, which means it proves "each L-shaped area is $k^3$" by the same proof that $2 times sum_{j=0}^k j = (0+k) + (1+k-1) + ... + (k-1+1) + (k+0) = (k+1) times k$, which makes it really close to the coloured rectangles above and below.
                                $endgroup$
                                – Wok
                                Sep 4 '11 at 7:49




                                $begingroup$
                                However, each L-shape coefficient has the same factor $k$, which means it proves "each L-shaped area is $k^3$" by the same proof that $2 times sum_{j=0}^k j = (0+k) + (1+k-1) + ... + (k-1+1) + (k+0) = (k+1) times k$, which makes it really close to the coloured rectangles above and below.
                                $endgroup$
                                – Wok
                                Sep 4 '11 at 7:49












                                $begingroup$
                                I hope you don't mind if I use both ideas in another answer.
                                $endgroup$
                                – Wok
                                Sep 4 '11 at 8:13




                                $begingroup$
                                I hope you don't mind if I use both ideas in another answer.
                                $endgroup$
                                – Wok
                                Sep 4 '11 at 8:13












                                $begingroup$
                                I don't mind. I simply find it less aesthetic to need to use $sumlimits_{j=1}^k;j=k(k+1)/2$ or that $(k(k+1)/2)^2-(k(k-1)/2)^2=k^3$ in an intuitive proof.
                                $endgroup$
                                – robjohn
                                Sep 4 '11 at 19:31






                                $begingroup$
                                I don't mind. I simply find it less aesthetic to need to use $sumlimits_{j=1}^k;j=k(k+1)/2$ or that $(k(k+1)/2)^2-(k(k-1)/2)^2=k^3$ in an intuitive proof.
                                $endgroup$
                                – robjohn
                                Sep 4 '11 at 19:31













                                32












                                $begingroup$

                                Can you get the intuition explanation from the following two pictures?[EDIT: the following is essentially the same as Mariano's answer. He didn't mentioned the first picture though.]



                                enter image description hereenter image description here



                                The images are from Brian R Sears.






                                share|cite|improve this answer











                                $endgroup$









                                • 1




                                  $begingroup$
                                  nice, but already posted (linked) by Mariano
                                  $endgroup$
                                  – leonbloy
                                  Sep 2 '11 at 23:24
















                                32












                                $begingroup$

                                Can you get the intuition explanation from the following two pictures?[EDIT: the following is essentially the same as Mariano's answer. He didn't mentioned the first picture though.]



                                enter image description hereenter image description here



                                The images are from Brian R Sears.






                                share|cite|improve this answer











                                $endgroup$









                                • 1




                                  $begingroup$
                                  nice, but already posted (linked) by Mariano
                                  $endgroup$
                                  – leonbloy
                                  Sep 2 '11 at 23:24














                                32












                                32








                                32





                                $begingroup$

                                Can you get the intuition explanation from the following two pictures?[EDIT: the following is essentially the same as Mariano's answer. He didn't mentioned the first picture though.]



                                enter image description hereenter image description here



                                The images are from Brian R Sears.






                                share|cite|improve this answer











                                $endgroup$



                                Can you get the intuition explanation from the following two pictures?[EDIT: the following is essentially the same as Mariano's answer. He didn't mentioned the first picture though.]



                                enter image description hereenter image description here



                                The images are from Brian R Sears.







                                share|cite|improve this answer














                                share|cite|improve this answer



                                share|cite|improve this answer








                                edited Sep 2 '11 at 23:27


























                                community wiki





                                2 revs
                                Jack









                                • 1




                                  $begingroup$
                                  nice, but already posted (linked) by Mariano
                                  $endgroup$
                                  – leonbloy
                                  Sep 2 '11 at 23:24














                                • 1




                                  $begingroup$
                                  nice, but already posted (linked) by Mariano
                                  $endgroup$
                                  – leonbloy
                                  Sep 2 '11 at 23:24








                                1




                                1




                                $begingroup$
                                nice, but already posted (linked) by Mariano
                                $endgroup$
                                – leonbloy
                                Sep 2 '11 at 23:24




                                $begingroup$
                                nice, but already posted (linked) by Mariano
                                $endgroup$
                                – leonbloy
                                Sep 2 '11 at 23:24











                                28












                                $begingroup$

                                I believe this illustration is due to Anders Kaseorg:



                                Visual proof by block-stacking






                                share|cite|improve this answer











                                $endgroup$


















                                  28












                                  $begingroup$

                                  I believe this illustration is due to Anders Kaseorg:



                                  Visual proof by block-stacking






                                  share|cite|improve this answer











                                  $endgroup$
















                                    28












                                    28








                                    28





                                    $begingroup$

                                    I believe this illustration is due to Anders Kaseorg:



                                    Visual proof by block-stacking






                                    share|cite|improve this answer











                                    $endgroup$



                                    I believe this illustration is due to Anders Kaseorg:



                                    Visual proof by block-stacking







                                    share|cite|improve this answer














                                    share|cite|improve this answer



                                    share|cite|improve this answer








                                    answered Apr 25 '13 at 2:17


























                                    community wiki





                                    MJD
























                                        23












                                        $begingroup$

                                        There's this nice picture from the Wikipedia entry on the squared triangular number:



                                        enter image description here



                                        The left side shows that $1 + 2 + 3$ forms a triangle and so that squaring it produces a larger triangle made up of $1+2+3$ copies of the original triangle. The right side has $1(1^2) + 2(2^2) + 3(3^2) = 1^3 + 2^3 + 3^3$. The coloring shows why the two sides are equal.



                                        There are several other references for combinatorial proofs and geometric arguments on the Wikipedia page.






                                        share|cite|improve this answer











                                        $endgroup$


















                                          23












                                          $begingroup$

                                          There's this nice picture from the Wikipedia entry on the squared triangular number:



                                          enter image description here



                                          The left side shows that $1 + 2 + 3$ forms a triangle and so that squaring it produces a larger triangle made up of $1+2+3$ copies of the original triangle. The right side has $1(1^2) + 2(2^2) + 3(3^2) = 1^3 + 2^3 + 3^3$. The coloring shows why the two sides are equal.



                                          There are several other references for combinatorial proofs and geometric arguments on the Wikipedia page.






                                          share|cite|improve this answer











                                          $endgroup$
















                                            23












                                            23








                                            23





                                            $begingroup$

                                            There's this nice picture from the Wikipedia entry on the squared triangular number:



                                            enter image description here



                                            The left side shows that $1 + 2 + 3$ forms a triangle and so that squaring it produces a larger triangle made up of $1+2+3$ copies of the original triangle. The right side has $1(1^2) + 2(2^2) + 3(3^2) = 1^3 + 2^3 + 3^3$. The coloring shows why the two sides are equal.



                                            There are several other references for combinatorial proofs and geometric arguments on the Wikipedia page.






                                            share|cite|improve this answer











                                            $endgroup$



                                            There's this nice picture from the Wikipedia entry on the squared triangular number:



                                            enter image description here



                                            The left side shows that $1 + 2 + 3$ forms a triangle and so that squaring it produces a larger triangle made up of $1+2+3$ copies of the original triangle. The right side has $1(1^2) + 2(2^2) + 3(3^2) = 1^3 + 2^3 + 3^3$. The coloring shows why the two sides are equal.



                                            There are several other references for combinatorial proofs and geometric arguments on the Wikipedia page.







                                            share|cite|improve this answer














                                            share|cite|improve this answer



                                            share|cite|improve this answer








                                            edited Sep 2 '11 at 22:15

























                                            answered Sep 2 '11 at 21:17









                                            Mike SpiveyMike Spivey

                                            42.8k8144234




                                            42.8k8144234























                                                21












                                                $begingroup$

                                                Here's another version of this "proof without words". This is the case $n=4$.



                                                enter image description here



                                                There are 1 $1 times 1$, 2 $2 times 2$, 3 $3 times 3$, ... squares, for a total area of $1^3 + 2^3 + ldots + n^3$. For even $k$, two of the $k times k$ squares overlap in a $k/2 times k/2$ square, but this
                                                just balances out a $k/2 times k/2$ square that is left out, so the total is the area of
                                                a square of side $1 + 2 + ldots + n$.






                                                share|cite|improve this answer









                                                $endgroup$


















                                                  21












                                                  $begingroup$

                                                  Here's another version of this "proof without words". This is the case $n=4$.



                                                  enter image description here



                                                  There are 1 $1 times 1$, 2 $2 times 2$, 3 $3 times 3$, ... squares, for a total area of $1^3 + 2^3 + ldots + n^3$. For even $k$, two of the $k times k$ squares overlap in a $k/2 times k/2$ square, but this
                                                  just balances out a $k/2 times k/2$ square that is left out, so the total is the area of
                                                  a square of side $1 + 2 + ldots + n$.






                                                  share|cite|improve this answer









                                                  $endgroup$
















                                                    21












                                                    21








                                                    21





                                                    $begingroup$

                                                    Here's another version of this "proof without words". This is the case $n=4$.



                                                    enter image description here



                                                    There are 1 $1 times 1$, 2 $2 times 2$, 3 $3 times 3$, ... squares, for a total area of $1^3 + 2^3 + ldots + n^3$. For even $k$, two of the $k times k$ squares overlap in a $k/2 times k/2$ square, but this
                                                    just balances out a $k/2 times k/2$ square that is left out, so the total is the area of
                                                    a square of side $1 + 2 + ldots + n$.






                                                    share|cite|improve this answer









                                                    $endgroup$



                                                    Here's another version of this "proof without words". This is the case $n=4$.



                                                    enter image description here



                                                    There are 1 $1 times 1$, 2 $2 times 2$, 3 $3 times 3$, ... squares, for a total area of $1^3 + 2^3 + ldots + n^3$. For even $k$, two of the $k times k$ squares overlap in a $k/2 times k/2$ square, but this
                                                    just balances out a $k/2 times k/2$ square that is left out, so the total is the area of
                                                    a square of side $1 + 2 + ldots + n$.







                                                    share|cite|improve this answer












                                                    share|cite|improve this answer



                                                    share|cite|improve this answer










                                                    answered Sep 2 '11 at 22:34









                                                    Robert IsraelRobert Israel

                                                    328k23216469




                                                    328k23216469























                                                        20












                                                        $begingroup$

                                                        Each colored area is $k^3$ as a difference of two areas: $S_k^2 - S_{k-1}^2$.



                                                        enter image description here



                                                        enter image description here





                                                        The detailed proof which comes with the drawing is the following.



                                                        For any positive integer $k$, we define:
                                                        $$S_i = sum_{j=1}^{i} j$$



                                                        We first notice:
                                                        $$S_i^2 = S_i^2 - S_0^2= sum_{k=1}^{i} left(S_k^2 - S_{k-1}^2right)$$



                                                        The expected result finally comes from:
                                                        $$S_k^2 - S_{k-1}^2 = k left(k+2 S_{k-1}right) = kleft(k+kleft(k-1right)right)=k^3$$






                                                        share|cite|improve this answer











                                                        $endgroup$









                                                        • 1




                                                          $begingroup$
                                                          So essentially, you are using the fact that $$left(sum_{j=1}^k;jright)^2-left(sum_{j=1}^{k-1};jright)^2=k^3$$ to justify the diagram which is supposed to prove that fact intuitively.
                                                          $endgroup$
                                                          – robjohn
                                                          Sep 4 '11 at 1:09












                                                        • $begingroup$
                                                          As you mentioned in another answer, this is where the diagram is the least intuitive. However, if you cannot picture $k^3$ on a 2D-plane, then you need another representation as a difference of areas.
                                                          $endgroup$
                                                          – Wok
                                                          Sep 4 '11 at 6:50










                                                        • $begingroup$
                                                          As soon as you know $S_k = sum_{j=1}^k j = frac{k(k+1)}{2}$, I find it more intuitive to figure out each colored area is $k^3$ on this diagram than to figure it out by counting squares plus two rectangles when $k$ is even.
                                                          $endgroup$
                                                          – Wok
                                                          Sep 4 '11 at 7:06


















                                                        20












                                                        $begingroup$

                                                        Each colored area is $k^3$ as a difference of two areas: $S_k^2 - S_{k-1}^2$.



                                                        enter image description here



                                                        enter image description here





                                                        The detailed proof which comes with the drawing is the following.



                                                        For any positive integer $k$, we define:
                                                        $$S_i = sum_{j=1}^{i} j$$



                                                        We first notice:
                                                        $$S_i^2 = S_i^2 - S_0^2= sum_{k=1}^{i} left(S_k^2 - S_{k-1}^2right)$$



                                                        The expected result finally comes from:
                                                        $$S_k^2 - S_{k-1}^2 = k left(k+2 S_{k-1}right) = kleft(k+kleft(k-1right)right)=k^3$$






                                                        share|cite|improve this answer











                                                        $endgroup$









                                                        • 1




                                                          $begingroup$
                                                          So essentially, you are using the fact that $$left(sum_{j=1}^k;jright)^2-left(sum_{j=1}^{k-1};jright)^2=k^3$$ to justify the diagram which is supposed to prove that fact intuitively.
                                                          $endgroup$
                                                          – robjohn
                                                          Sep 4 '11 at 1:09












                                                        • $begingroup$
                                                          As you mentioned in another answer, this is where the diagram is the least intuitive. However, if you cannot picture $k^3$ on a 2D-plane, then you need another representation as a difference of areas.
                                                          $endgroup$
                                                          – Wok
                                                          Sep 4 '11 at 6:50










                                                        • $begingroup$
                                                          As soon as you know $S_k = sum_{j=1}^k j = frac{k(k+1)}{2}$, I find it more intuitive to figure out each colored area is $k^3$ on this diagram than to figure it out by counting squares plus two rectangles when $k$ is even.
                                                          $endgroup$
                                                          – Wok
                                                          Sep 4 '11 at 7:06
















                                                        20












                                                        20








                                                        20





                                                        $begingroup$

                                                        Each colored area is $k^3$ as a difference of two areas: $S_k^2 - S_{k-1}^2$.



                                                        enter image description here



                                                        enter image description here





                                                        The detailed proof which comes with the drawing is the following.



                                                        For any positive integer $k$, we define:
                                                        $$S_i = sum_{j=1}^{i} j$$



                                                        We first notice:
                                                        $$S_i^2 = S_i^2 - S_0^2= sum_{k=1}^{i} left(S_k^2 - S_{k-1}^2right)$$



                                                        The expected result finally comes from:
                                                        $$S_k^2 - S_{k-1}^2 = k left(k+2 S_{k-1}right) = kleft(k+kleft(k-1right)right)=k^3$$






                                                        share|cite|improve this answer











                                                        $endgroup$



                                                        Each colored area is $k^3$ as a difference of two areas: $S_k^2 - S_{k-1}^2$.



                                                        enter image description here



                                                        enter image description here





                                                        The detailed proof which comes with the drawing is the following.



                                                        For any positive integer $k$, we define:
                                                        $$S_i = sum_{j=1}^{i} j$$



                                                        We first notice:
                                                        $$S_i^2 = S_i^2 - S_0^2= sum_{k=1}^{i} left(S_k^2 - S_{k-1}^2right)$$



                                                        The expected result finally comes from:
                                                        $$S_k^2 - S_{k-1}^2 = k left(k+2 S_{k-1}right) = kleft(k+kleft(k-1right)right)=k^3$$







                                                        share|cite|improve this answer














                                                        share|cite|improve this answer



                                                        share|cite|improve this answer








                                                        edited Sep 3 '11 at 8:07

























                                                        answered Sep 3 '11 at 7:18









                                                        WokWok

                                                        1,5781323




                                                        1,5781323








                                                        • 1




                                                          $begingroup$
                                                          So essentially, you are using the fact that $$left(sum_{j=1}^k;jright)^2-left(sum_{j=1}^{k-1};jright)^2=k^3$$ to justify the diagram which is supposed to prove that fact intuitively.
                                                          $endgroup$
                                                          – robjohn
                                                          Sep 4 '11 at 1:09












                                                        • $begingroup$
                                                          As you mentioned in another answer, this is where the diagram is the least intuitive. However, if you cannot picture $k^3$ on a 2D-plane, then you need another representation as a difference of areas.
                                                          $endgroup$
                                                          – Wok
                                                          Sep 4 '11 at 6:50










                                                        • $begingroup$
                                                          As soon as you know $S_k = sum_{j=1}^k j = frac{k(k+1)}{2}$, I find it more intuitive to figure out each colored area is $k^3$ on this diagram than to figure it out by counting squares plus two rectangles when $k$ is even.
                                                          $endgroup$
                                                          – Wok
                                                          Sep 4 '11 at 7:06
















                                                        • 1




                                                          $begingroup$
                                                          So essentially, you are using the fact that $$left(sum_{j=1}^k;jright)^2-left(sum_{j=1}^{k-1};jright)^2=k^3$$ to justify the diagram which is supposed to prove that fact intuitively.
                                                          $endgroup$
                                                          – robjohn
                                                          Sep 4 '11 at 1:09












                                                        • $begingroup$
                                                          As you mentioned in another answer, this is where the diagram is the least intuitive. However, if you cannot picture $k^3$ on a 2D-plane, then you need another representation as a difference of areas.
                                                          $endgroup$
                                                          – Wok
                                                          Sep 4 '11 at 6:50










                                                        • $begingroup$
                                                          As soon as you know $S_k = sum_{j=1}^k j = frac{k(k+1)}{2}$, I find it more intuitive to figure out each colored area is $k^3$ on this diagram than to figure it out by counting squares plus two rectangles when $k$ is even.
                                                          $endgroup$
                                                          – Wok
                                                          Sep 4 '11 at 7:06










                                                        1




                                                        1




                                                        $begingroup$
                                                        So essentially, you are using the fact that $$left(sum_{j=1}^k;jright)^2-left(sum_{j=1}^{k-1};jright)^2=k^3$$ to justify the diagram which is supposed to prove that fact intuitively.
                                                        $endgroup$
                                                        – robjohn
                                                        Sep 4 '11 at 1:09






                                                        $begingroup$
                                                        So essentially, you are using the fact that $$left(sum_{j=1}^k;jright)^2-left(sum_{j=1}^{k-1};jright)^2=k^3$$ to justify the diagram which is supposed to prove that fact intuitively.
                                                        $endgroup$
                                                        – robjohn
                                                        Sep 4 '11 at 1:09














                                                        $begingroup$
                                                        As you mentioned in another answer, this is where the diagram is the least intuitive. However, if you cannot picture $k^3$ on a 2D-plane, then you need another representation as a difference of areas.
                                                        $endgroup$
                                                        – Wok
                                                        Sep 4 '11 at 6:50




                                                        $begingroup$
                                                        As you mentioned in another answer, this is where the diagram is the least intuitive. However, if you cannot picture $k^3$ on a 2D-plane, then you need another representation as a difference of areas.
                                                        $endgroup$
                                                        – Wok
                                                        Sep 4 '11 at 6:50












                                                        $begingroup$
                                                        As soon as you know $S_k = sum_{j=1}^k j = frac{k(k+1)}{2}$, I find it more intuitive to figure out each colored area is $k^3$ on this diagram than to figure it out by counting squares plus two rectangles when $k$ is even.
                                                        $endgroup$
                                                        – Wok
                                                        Sep 4 '11 at 7:06






                                                        $begingroup$
                                                        As soon as you know $S_k = sum_{j=1}^k j = frac{k(k+1)}{2}$, I find it more intuitive to figure out each colored area is $k^3$ on this diagram than to figure it out by counting squares plus two rectangles when $k$ is even.
                                                        $endgroup$
                                                        – Wok
                                                        Sep 4 '11 at 7:06













                                                        17












                                                        $begingroup$

                                                        The formula is due to Nicomachus of Gerasa. There is a nice discussion of ways to prove it at this n-category cafe post, including a bijective proof and some visual / "geometric" proofs.






                                                        share|cite|improve this answer









                                                        $endgroup$


















                                                          17












                                                          $begingroup$

                                                          The formula is due to Nicomachus of Gerasa. There is a nice discussion of ways to prove it at this n-category cafe post, including a bijective proof and some visual / "geometric" proofs.






                                                          share|cite|improve this answer









                                                          $endgroup$
















                                                            17












                                                            17








                                                            17





                                                            $begingroup$

                                                            The formula is due to Nicomachus of Gerasa. There is a nice discussion of ways to prove it at this n-category cafe post, including a bijective proof and some visual / "geometric" proofs.






                                                            share|cite|improve this answer









                                                            $endgroup$



                                                            The formula is due to Nicomachus of Gerasa. There is a nice discussion of ways to prove it at this n-category cafe post, including a bijective proof and some visual / "geometric" proofs.







                                                            share|cite|improve this answer












                                                            share|cite|improve this answer



                                                            share|cite|improve this answer










                                                            answered Jan 22 '11 at 19:12









                                                            Qiaochu YuanQiaochu Yuan

                                                            281k32593938




                                                            281k32593938























                                                                17












                                                                $begingroup$

                                                                Several visual proofs of this indentity are collected in the book
                                                                Roger B. Nelsen: Proofs without Words
                                                                starting from p.84.



                                                                Although several of these proofs can still be considered inductive, I thought it might be interesting to mention them.



                                                                Original sources are given on p. 147:




                                                                • 84 Mathematical Gazette, vol. 49, no. 368 (May 1965), p. 199. jstor

                                                                • 85 Mathematics Magazine, vol. 50, no. 2 (March 1977), p. 74. jstor

                                                                • 86 Mathematics Magazine, vol. 58, no. 1 (Jan. 1985), p. 11. jstor

                                                                • 87 Mathematics Magazine, vol. 62, no. 4 (Oct. 1989), p. 259. jstor

                                                                • 87 Mathematical Gazette, vol. 49, no. 368 (May 1965), p. 200. jstor

                                                                • 88 Mathematics Magazine, vol. 63, no. 3 (June 1990), p. 178. jstor

                                                                • 89 Mathematics Magazine, vol. 62, no. 5 (Dec. 1989), p. 323. jstor

                                                                • 90 Mathematics Magazine, vol. 65, no. 3 (June 1992), p. 185. jstor






                                                                share|cite|improve this answer











                                                                $endgroup$













                                                                • $begingroup$
                                                                  Great collections. Thanks
                                                                  $endgroup$
                                                                  – mrs
                                                                  Jun 27 '12 at 11:11
















                                                                17












                                                                $begingroup$

                                                                Several visual proofs of this indentity are collected in the book
                                                                Roger B. Nelsen: Proofs without Words
                                                                starting from p.84.



                                                                Although several of these proofs can still be considered inductive, I thought it might be interesting to mention them.



                                                                Original sources are given on p. 147:




                                                                • 84 Mathematical Gazette, vol. 49, no. 368 (May 1965), p. 199. jstor

                                                                • 85 Mathematics Magazine, vol. 50, no. 2 (March 1977), p. 74. jstor

                                                                • 86 Mathematics Magazine, vol. 58, no. 1 (Jan. 1985), p. 11. jstor

                                                                • 87 Mathematics Magazine, vol. 62, no. 4 (Oct. 1989), p. 259. jstor

                                                                • 87 Mathematical Gazette, vol. 49, no. 368 (May 1965), p. 200. jstor

                                                                • 88 Mathematics Magazine, vol. 63, no. 3 (June 1990), p. 178. jstor

                                                                • 89 Mathematics Magazine, vol. 62, no. 5 (Dec. 1989), p. 323. jstor

                                                                • 90 Mathematics Magazine, vol. 65, no. 3 (June 1992), p. 185. jstor






                                                                share|cite|improve this answer











                                                                $endgroup$













                                                                • $begingroup$
                                                                  Great collections. Thanks
                                                                  $endgroup$
                                                                  – mrs
                                                                  Jun 27 '12 at 11:11














                                                                17












                                                                17








                                                                17





                                                                $begingroup$

                                                                Several visual proofs of this indentity are collected in the book
                                                                Roger B. Nelsen: Proofs without Words
                                                                starting from p.84.



                                                                Although several of these proofs can still be considered inductive, I thought it might be interesting to mention them.



                                                                Original sources are given on p. 147:




                                                                • 84 Mathematical Gazette, vol. 49, no. 368 (May 1965), p. 199. jstor

                                                                • 85 Mathematics Magazine, vol. 50, no. 2 (March 1977), p. 74. jstor

                                                                • 86 Mathematics Magazine, vol. 58, no. 1 (Jan. 1985), p. 11. jstor

                                                                • 87 Mathematics Magazine, vol. 62, no. 4 (Oct. 1989), p. 259. jstor

                                                                • 87 Mathematical Gazette, vol. 49, no. 368 (May 1965), p. 200. jstor

                                                                • 88 Mathematics Magazine, vol. 63, no. 3 (June 1990), p. 178. jstor

                                                                • 89 Mathematics Magazine, vol. 62, no. 5 (Dec. 1989), p. 323. jstor

                                                                • 90 Mathematics Magazine, vol. 65, no. 3 (June 1992), p. 185. jstor






                                                                share|cite|improve this answer











                                                                $endgroup$



                                                                Several visual proofs of this indentity are collected in the book
                                                                Roger B. Nelsen: Proofs without Words
                                                                starting from p.84.



                                                                Although several of these proofs can still be considered inductive, I thought it might be interesting to mention them.



                                                                Original sources are given on p. 147:




                                                                • 84 Mathematical Gazette, vol. 49, no. 368 (May 1965), p. 199. jstor

                                                                • 85 Mathematics Magazine, vol. 50, no. 2 (March 1977), p. 74. jstor

                                                                • 86 Mathematics Magazine, vol. 58, no. 1 (Jan. 1985), p. 11. jstor

                                                                • 87 Mathematics Magazine, vol. 62, no. 4 (Oct. 1989), p. 259. jstor

                                                                • 87 Mathematical Gazette, vol. 49, no. 368 (May 1965), p. 200. jstor

                                                                • 88 Mathematics Magazine, vol. 63, no. 3 (June 1990), p. 178. jstor

                                                                • 89 Mathematics Magazine, vol. 62, no. 5 (Dec. 1989), p. 323. jstor

                                                                • 90 Mathematics Magazine, vol. 65, no. 3 (June 1992), p. 185. jstor







                                                                share|cite|improve this answer














                                                                share|cite|improve this answer



                                                                share|cite|improve this answer








                                                                answered Jun 27 '12 at 11:00


























                                                                community wiki





                                                                Martin Sleziak













                                                                • $begingroup$
                                                                  Great collections. Thanks
                                                                  $endgroup$
                                                                  – mrs
                                                                  Jun 27 '12 at 11:11


















                                                                • $begingroup$
                                                                  Great collections. Thanks
                                                                  $endgroup$
                                                                  – mrs
                                                                  Jun 27 '12 at 11:11
















                                                                $begingroup$
                                                                Great collections. Thanks
                                                                $endgroup$
                                                                – mrs
                                                                Jun 27 '12 at 11:11




                                                                $begingroup$
                                                                Great collections. Thanks
                                                                $endgroup$
                                                                – mrs
                                                                Jun 27 '12 at 11:11











                                                                11












                                                                $begingroup$

                                                                The sum of a degree $n$ polynomial $f(n)$ will be a degree $n+1$ polynomial $S(n)$ for $n geq 0$ and both polynomials can be extended (maintaining the relation $S(n)-S(n-1) = f(n)$) to negative $n$.
                                                                To verify that the formula for $Sigma k^3$ is correct one need only test it for any 5 distinct values of $n$, but the structure of the answer can be predicted algebraically using the continuation to negative $n$.



                                                                If $S(n) = (1^3 + 2^3 + dots n^3)$ is the polynomial that satisfies $S(n)-S(n-1) = n^3$ and $S(1)=1$, then one can calculate from that equation that $S(0)=S(-1)=0$ and $S(-n-1)=S(n)$ for all negative $n$, so that $S$ is symmetric around $-1/2$. The vanishing at 0 and -1 implies that $S(t)$ is divisible as a polynomial by $t(t+1)$. The symmetry implies that $S(t)$ is a function (necessarily a polynomial) of $t(t+1)$.



                                                                $S(t)$ being of degree 4, this means $S(n) = a (n)(n+1) + b((n^2 +n)^2$ for constants $a$ and $b$. Summation being analogous to integration (and equal to it in a suitable limit), they have to agree on highest degree terms. Here it forces $b$ to be $1/4$ to match $int x^3 = x^4/4$. Computing the sum at a single point such as $n=1$ determines $a$, which is zero.



                                                                Similar reasoning shows that $S_k(n)$ is divisible as a polynomial by $n(n+1)$ for all $k$. For odd $k$, $S_k(n)$ is a polynomial in $n(n+1)$.






                                                                share|cite|improve this answer











                                                                $endgroup$


















                                                                  11












                                                                  $begingroup$

                                                                  The sum of a degree $n$ polynomial $f(n)$ will be a degree $n+1$ polynomial $S(n)$ for $n geq 0$ and both polynomials can be extended (maintaining the relation $S(n)-S(n-1) = f(n)$) to negative $n$.
                                                                  To verify that the formula for $Sigma k^3$ is correct one need only test it for any 5 distinct values of $n$, but the structure of the answer can be predicted algebraically using the continuation to negative $n$.



                                                                  If $S(n) = (1^3 + 2^3 + dots n^3)$ is the polynomial that satisfies $S(n)-S(n-1) = n^3$ and $S(1)=1$, then one can calculate from that equation that $S(0)=S(-1)=0$ and $S(-n-1)=S(n)$ for all negative $n$, so that $S$ is symmetric around $-1/2$. The vanishing at 0 and -1 implies that $S(t)$ is divisible as a polynomial by $t(t+1)$. The symmetry implies that $S(t)$ is a function (necessarily a polynomial) of $t(t+1)$.



                                                                  $S(t)$ being of degree 4, this means $S(n) = a (n)(n+1) + b((n^2 +n)^2$ for constants $a$ and $b$. Summation being analogous to integration (and equal to it in a suitable limit), they have to agree on highest degree terms. Here it forces $b$ to be $1/4$ to match $int x^3 = x^4/4$. Computing the sum at a single point such as $n=1$ determines $a$, which is zero.



                                                                  Similar reasoning shows that $S_k(n)$ is divisible as a polynomial by $n(n+1)$ for all $k$. For odd $k$, $S_k(n)$ is a polynomial in $n(n+1)$.






                                                                  share|cite|improve this answer











                                                                  $endgroup$
















                                                                    11












                                                                    11








                                                                    11





                                                                    $begingroup$

                                                                    The sum of a degree $n$ polynomial $f(n)$ will be a degree $n+1$ polynomial $S(n)$ for $n geq 0$ and both polynomials can be extended (maintaining the relation $S(n)-S(n-1) = f(n)$) to negative $n$.
                                                                    To verify that the formula for $Sigma k^3$ is correct one need only test it for any 5 distinct values of $n$, but the structure of the answer can be predicted algebraically using the continuation to negative $n$.



                                                                    If $S(n) = (1^3 + 2^3 + dots n^3)$ is the polynomial that satisfies $S(n)-S(n-1) = n^3$ and $S(1)=1$, then one can calculate from that equation that $S(0)=S(-1)=0$ and $S(-n-1)=S(n)$ for all negative $n$, so that $S$ is symmetric around $-1/2$. The vanishing at 0 and -1 implies that $S(t)$ is divisible as a polynomial by $t(t+1)$. The symmetry implies that $S(t)$ is a function (necessarily a polynomial) of $t(t+1)$.



                                                                    $S(t)$ being of degree 4, this means $S(n) = a (n)(n+1) + b((n^2 +n)^2$ for constants $a$ and $b$. Summation being analogous to integration (and equal to it in a suitable limit), they have to agree on highest degree terms. Here it forces $b$ to be $1/4$ to match $int x^3 = x^4/4$. Computing the sum at a single point such as $n=1$ determines $a$, which is zero.



                                                                    Similar reasoning shows that $S_k(n)$ is divisible as a polynomial by $n(n+1)$ for all $k$. For odd $k$, $S_k(n)$ is a polynomial in $n(n+1)$.






                                                                    share|cite|improve this answer











                                                                    $endgroup$



                                                                    The sum of a degree $n$ polynomial $f(n)$ will be a degree $n+1$ polynomial $S(n)$ for $n geq 0$ and both polynomials can be extended (maintaining the relation $S(n)-S(n-1) = f(n)$) to negative $n$.
                                                                    To verify that the formula for $Sigma k^3$ is correct one need only test it for any 5 distinct values of $n$, but the structure of the answer can be predicted algebraically using the continuation to negative $n$.



                                                                    If $S(n) = (1^3 + 2^3 + dots n^3)$ is the polynomial that satisfies $S(n)-S(n-1) = n^3$ and $S(1)=1$, then one can calculate from that equation that $S(0)=S(-1)=0$ and $S(-n-1)=S(n)$ for all negative $n$, so that $S$ is symmetric around $-1/2$. The vanishing at 0 and -1 implies that $S(t)$ is divisible as a polynomial by $t(t+1)$. The symmetry implies that $S(t)$ is a function (necessarily a polynomial) of $t(t+1)$.



                                                                    $S(t)$ being of degree 4, this means $S(n) = a (n)(n+1) + b((n^2 +n)^2$ for constants $a$ and $b$. Summation being analogous to integration (and equal to it in a suitable limit), they have to agree on highest degree terms. Here it forces $b$ to be $1/4$ to match $int x^3 = x^4/4$. Computing the sum at a single point such as $n=1$ determines $a$, which is zero.



                                                                    Similar reasoning shows that $S_k(n)$ is divisible as a polynomial by $n(n+1)$ for all $k$. For odd $k$, $S_k(n)$ is a polynomial in $n(n+1)$.







                                                                    share|cite|improve this answer














                                                                    share|cite|improve this answer



                                                                    share|cite|improve this answer








                                                                    edited Nov 1 '11 at 6:31

























                                                                    answered Sep 2 '11 at 23:38









                                                                    zyxzyx

                                                                    31.6k33699




                                                                    31.6k33699























                                                                        11












                                                                        $begingroup$

                                                                        Here's a direct algebraic proof. $$sum_{k=1}^n(k^3-k^2)=2sum_{k=1}^nkcdotfrac{k(k-1)}2=2sum_{k=1}^nksum_{l=1}^{k-1}l=2sum_{1leqslant l<kleqslant n}kl=left(sum_{k=1}^nkright)^2-sum_{k=1}^nk^2$$






                                                                        share|cite|improve this answer









                                                                        $endgroup$


















                                                                          11












                                                                          $begingroup$

                                                                          Here's a direct algebraic proof. $$sum_{k=1}^n(k^3-k^2)=2sum_{k=1}^nkcdotfrac{k(k-1)}2=2sum_{k=1}^nksum_{l=1}^{k-1}l=2sum_{1leqslant l<kleqslant n}kl=left(sum_{k=1}^nkright)^2-sum_{k=1}^nk^2$$






                                                                          share|cite|improve this answer









                                                                          $endgroup$
















                                                                            11












                                                                            11








                                                                            11





                                                                            $begingroup$

                                                                            Here's a direct algebraic proof. $$sum_{k=1}^n(k^3-k^2)=2sum_{k=1}^nkcdotfrac{k(k-1)}2=2sum_{k=1}^nksum_{l=1}^{k-1}l=2sum_{1leqslant l<kleqslant n}kl=left(sum_{k=1}^nkright)^2-sum_{k=1}^nk^2$$






                                                                            share|cite|improve this answer









                                                                            $endgroup$



                                                                            Here's a direct algebraic proof. $$sum_{k=1}^n(k^3-k^2)=2sum_{k=1}^nkcdotfrac{k(k-1)}2=2sum_{k=1}^nksum_{l=1}^{k-1}l=2sum_{1leqslant l<kleqslant n}kl=left(sum_{k=1}^nkright)^2-sum_{k=1}^nk^2$$







                                                                            share|cite|improve this answer












                                                                            share|cite|improve this answer



                                                                            share|cite|improve this answer










                                                                            answered Apr 1 '15 at 13:27









                                                                            rabotarabota

                                                                            14.3k32782




                                                                            14.3k32782























                                                                                10












                                                                                $begingroup$

                                                                                You know, $sum_0^n x^k=frac{1-x^{n+1}}{1-x}$. Differentiate both sides once, $sum_1^n kx^{k-1}=frac{x^n(nx-n-1)+1}{x^2-2x+1}$. Now taking $lim_{xto1}$ both sides and then squaring the result will give you the expression on the RHS. You can further differentiate $sum_0^n x^k=frac{1-x^{n+1}}{1-x}$ until you get $k^3$ inside the expression, take limit again you will get the same result as of $(lim_{xto1}frac{x^n(nx-n-1)+1}{x^2-2x+1})^2$. You can also prove it using telescopic series.






                                                                                share|cite|improve this answer











                                                                                $endgroup$









                                                                                • 1




                                                                                  $begingroup$
                                                                                  why does the assumption hold? this is usually proved using induction...
                                                                                  $endgroup$
                                                                                  – akkkk
                                                                                  Jun 27 '12 at 9:47










                                                                                • $begingroup$
                                                                                  what assumption??
                                                                                  $endgroup$
                                                                                  – Aang
                                                                                  Jun 27 '12 at 9:48










                                                                                • $begingroup$
                                                                                  I guess Auke means $sum_{0}^{n} x^k=frac{1-x^{n+1}}{1-x}$.
                                                                                  $endgroup$
                                                                                  – sdcvvc
                                                                                  Jun 27 '12 at 10:12












                                                                                • $begingroup$
                                                                                  LHS is a geometric series. en.wikipedia.org/wiki/Geometric_progression
                                                                                  $endgroup$
                                                                                  – Aang
                                                                                  Jun 27 '12 at 10:14








                                                                                • 1




                                                                                  $begingroup$
                                                                                  @Auke: one can also prove it like this - let $f(x) = sumlimits_{k=0}^nx^k$, then $f(x) - xf(x)$ is: $$ begin{align} 1+&x+x^2+dots+x^n \ &- \ &x+x^2+dots+x^n+x^{n+1} end{align} $$ which is $1-x^{n+1}$. Hence $$ (1-x)f(x) = 1-x^{n+1} $$ as needed.
                                                                                  $endgroup$
                                                                                  – Ilya
                                                                                  Jun 27 '12 at 12:40


















                                                                                10












                                                                                $begingroup$

                                                                                You know, $sum_0^n x^k=frac{1-x^{n+1}}{1-x}$. Differentiate both sides once, $sum_1^n kx^{k-1}=frac{x^n(nx-n-1)+1}{x^2-2x+1}$. Now taking $lim_{xto1}$ both sides and then squaring the result will give you the expression on the RHS. You can further differentiate $sum_0^n x^k=frac{1-x^{n+1}}{1-x}$ until you get $k^3$ inside the expression, take limit again you will get the same result as of $(lim_{xto1}frac{x^n(nx-n-1)+1}{x^2-2x+1})^2$. You can also prove it using telescopic series.






                                                                                share|cite|improve this answer











                                                                                $endgroup$









                                                                                • 1




                                                                                  $begingroup$
                                                                                  why does the assumption hold? this is usually proved using induction...
                                                                                  $endgroup$
                                                                                  – akkkk
                                                                                  Jun 27 '12 at 9:47










                                                                                • $begingroup$
                                                                                  what assumption??
                                                                                  $endgroup$
                                                                                  – Aang
                                                                                  Jun 27 '12 at 9:48










                                                                                • $begingroup$
                                                                                  I guess Auke means $sum_{0}^{n} x^k=frac{1-x^{n+1}}{1-x}$.
                                                                                  $endgroup$
                                                                                  – sdcvvc
                                                                                  Jun 27 '12 at 10:12












                                                                                • $begingroup$
                                                                                  LHS is a geometric series. en.wikipedia.org/wiki/Geometric_progression
                                                                                  $endgroup$
                                                                                  – Aang
                                                                                  Jun 27 '12 at 10:14








                                                                                • 1




                                                                                  $begingroup$
                                                                                  @Auke: one can also prove it like this - let $f(x) = sumlimits_{k=0}^nx^k$, then $f(x) - xf(x)$ is: $$ begin{align} 1+&x+x^2+dots+x^n \ &- \ &x+x^2+dots+x^n+x^{n+1} end{align} $$ which is $1-x^{n+1}$. Hence $$ (1-x)f(x) = 1-x^{n+1} $$ as needed.
                                                                                  $endgroup$
                                                                                  – Ilya
                                                                                  Jun 27 '12 at 12:40
















                                                                                10












                                                                                10








                                                                                10





                                                                                $begingroup$

                                                                                You know, $sum_0^n x^k=frac{1-x^{n+1}}{1-x}$. Differentiate both sides once, $sum_1^n kx^{k-1}=frac{x^n(nx-n-1)+1}{x^2-2x+1}$. Now taking $lim_{xto1}$ both sides and then squaring the result will give you the expression on the RHS. You can further differentiate $sum_0^n x^k=frac{1-x^{n+1}}{1-x}$ until you get $k^3$ inside the expression, take limit again you will get the same result as of $(lim_{xto1}frac{x^n(nx-n-1)+1}{x^2-2x+1})^2$. You can also prove it using telescopic series.






                                                                                share|cite|improve this answer











                                                                                $endgroup$



                                                                                You know, $sum_0^n x^k=frac{1-x^{n+1}}{1-x}$. Differentiate both sides once, $sum_1^n kx^{k-1}=frac{x^n(nx-n-1)+1}{x^2-2x+1}$. Now taking $lim_{xto1}$ both sides and then squaring the result will give you the expression on the RHS. You can further differentiate $sum_0^n x^k=frac{1-x^{n+1}}{1-x}$ until you get $k^3$ inside the expression, take limit again you will get the same result as of $(lim_{xto1}frac{x^n(nx-n-1)+1}{x^2-2x+1})^2$. You can also prove it using telescopic series.







                                                                                share|cite|improve this answer














                                                                                share|cite|improve this answer



                                                                                share|cite|improve this answer








                                                                                edited Jun 27 '12 at 9:40







                                                                                user5501

















                                                                                answered Jun 27 '12 at 9:29









                                                                                AangAang

                                                                                12.7k22365




                                                                                12.7k22365








                                                                                • 1




                                                                                  $begingroup$
                                                                                  why does the assumption hold? this is usually proved using induction...
                                                                                  $endgroup$
                                                                                  – akkkk
                                                                                  Jun 27 '12 at 9:47










                                                                                • $begingroup$
                                                                                  what assumption??
                                                                                  $endgroup$
                                                                                  – Aang
                                                                                  Jun 27 '12 at 9:48










                                                                                • $begingroup$
                                                                                  I guess Auke means $sum_{0}^{n} x^k=frac{1-x^{n+1}}{1-x}$.
                                                                                  $endgroup$
                                                                                  – sdcvvc
                                                                                  Jun 27 '12 at 10:12












                                                                                • $begingroup$
                                                                                  LHS is a geometric series. en.wikipedia.org/wiki/Geometric_progression
                                                                                  $endgroup$
                                                                                  – Aang
                                                                                  Jun 27 '12 at 10:14








                                                                                • 1




                                                                                  $begingroup$
                                                                                  @Auke: one can also prove it like this - let $f(x) = sumlimits_{k=0}^nx^k$, then $f(x) - xf(x)$ is: $$ begin{align} 1+&x+x^2+dots+x^n \ &- \ &x+x^2+dots+x^n+x^{n+1} end{align} $$ which is $1-x^{n+1}$. Hence $$ (1-x)f(x) = 1-x^{n+1} $$ as needed.
                                                                                  $endgroup$
                                                                                  – Ilya
                                                                                  Jun 27 '12 at 12:40
















                                                                                • 1




                                                                                  $begingroup$
                                                                                  why does the assumption hold? this is usually proved using induction...
                                                                                  $endgroup$
                                                                                  – akkkk
                                                                                  Jun 27 '12 at 9:47










                                                                                • $begingroup$
                                                                                  what assumption??
                                                                                  $endgroup$
                                                                                  – Aang
                                                                                  Jun 27 '12 at 9:48










                                                                                • $begingroup$
                                                                                  I guess Auke means $sum_{0}^{n} x^k=frac{1-x^{n+1}}{1-x}$.
                                                                                  $endgroup$
                                                                                  – sdcvvc
                                                                                  Jun 27 '12 at 10:12












                                                                                • $begingroup$
                                                                                  LHS is a geometric series. en.wikipedia.org/wiki/Geometric_progression
                                                                                  $endgroup$
                                                                                  – Aang
                                                                                  Jun 27 '12 at 10:14








                                                                                • 1




                                                                                  $begingroup$
                                                                                  @Auke: one can also prove it like this - let $f(x) = sumlimits_{k=0}^nx^k$, then $f(x) - xf(x)$ is: $$ begin{align} 1+&x+x^2+dots+x^n \ &- \ &x+x^2+dots+x^n+x^{n+1} end{align} $$ which is $1-x^{n+1}$. Hence $$ (1-x)f(x) = 1-x^{n+1} $$ as needed.
                                                                                  $endgroup$
                                                                                  – Ilya
                                                                                  Jun 27 '12 at 12:40










                                                                                1




                                                                                1




                                                                                $begingroup$
                                                                                why does the assumption hold? this is usually proved using induction...
                                                                                $endgroup$
                                                                                – akkkk
                                                                                Jun 27 '12 at 9:47




                                                                                $begingroup$
                                                                                why does the assumption hold? this is usually proved using induction...
                                                                                $endgroup$
                                                                                – akkkk
                                                                                Jun 27 '12 at 9:47












                                                                                $begingroup$
                                                                                what assumption??
                                                                                $endgroup$
                                                                                – Aang
                                                                                Jun 27 '12 at 9:48




                                                                                $begingroup$
                                                                                what assumption??
                                                                                $endgroup$
                                                                                – Aang
                                                                                Jun 27 '12 at 9:48












                                                                                $begingroup$
                                                                                I guess Auke means $sum_{0}^{n} x^k=frac{1-x^{n+1}}{1-x}$.
                                                                                $endgroup$
                                                                                – sdcvvc
                                                                                Jun 27 '12 at 10:12






                                                                                $begingroup$
                                                                                I guess Auke means $sum_{0}^{n} x^k=frac{1-x^{n+1}}{1-x}$.
                                                                                $endgroup$
                                                                                – sdcvvc
                                                                                Jun 27 '12 at 10:12














                                                                                $begingroup$
                                                                                LHS is a geometric series. en.wikipedia.org/wiki/Geometric_progression
                                                                                $endgroup$
                                                                                – Aang
                                                                                Jun 27 '12 at 10:14






                                                                                $begingroup$
                                                                                LHS is a geometric series. en.wikipedia.org/wiki/Geometric_progression
                                                                                $endgroup$
                                                                                – Aang
                                                                                Jun 27 '12 at 10:14






                                                                                1




                                                                                1




                                                                                $begingroup$
                                                                                @Auke: one can also prove it like this - let $f(x) = sumlimits_{k=0}^nx^k$, then $f(x) - xf(x)$ is: $$ begin{align} 1+&x+x^2+dots+x^n \ &- \ &x+x^2+dots+x^n+x^{n+1} end{align} $$ which is $1-x^{n+1}$. Hence $$ (1-x)f(x) = 1-x^{n+1} $$ as needed.
                                                                                $endgroup$
                                                                                – Ilya
                                                                                Jun 27 '12 at 12:40






                                                                                $begingroup$
                                                                                @Auke: one can also prove it like this - let $f(x) = sumlimits_{k=0}^nx^k$, then $f(x) - xf(x)$ is: $$ begin{align} 1+&x+x^2+dots+x^n \ &- \ &x+x^2+dots+x^n+x^{n+1} end{align} $$ which is $1-x^{n+1}$. Hence $$ (1-x)f(x) = 1-x^{n+1} $$ as needed.
                                                                                $endgroup$
                                                                                – Ilya
                                                                                Jun 27 '12 at 12:40













                                                                                10












                                                                                $begingroup$

                                                                                We know that $$A=sum_1^n k^3=frac{1}{4}n^4+frac{1}{2}n^3+frac{1}{4}n^2$$ and $$B=sum_1^n k=frac{1}{2}n^2+frac{1}{2}n$$ $A-B^2=0$. :)






                                                                                share|cite|improve this answer











                                                                                $endgroup$









                                                                                • 2




                                                                                  $begingroup$
                                                                                  If you are presenting this proof,write it nicely. $A=frac{n^2(n+1)^2}{4}$ and $B=frac{n(n+1)}{2}$,obviously $A=B^2$
                                                                                  $endgroup$
                                                                                  – Aang
                                                                                  Jun 27 '12 at 10:40










                                                                                • $begingroup$
                                                                                  @avatar: Dear Avatar; I will. Your proof is great and it looks analytically. Thanks. :-)
                                                                                  $endgroup$
                                                                                  – mrs
                                                                                  Jun 27 '12 at 10:48










                                                                                • $begingroup$
                                                                                  sorry, if you felt that i was being rude.
                                                                                  $endgroup$
                                                                                  – Aang
                                                                                  Jun 27 '12 at 10:53










                                                                                • $begingroup$
                                                                                  @avatar: Your proof lights mine. No Problem at all. WELCOME Avatar. :) :)
                                                                                  $endgroup$
                                                                                  – mrs
                                                                                  Jun 27 '12 at 10:57
















                                                                                10












                                                                                $begingroup$

                                                                                We know that $$A=sum_1^n k^3=frac{1}{4}n^4+frac{1}{2}n^3+frac{1}{4}n^2$$ and $$B=sum_1^n k=frac{1}{2}n^2+frac{1}{2}n$$ $A-B^2=0$. :)






                                                                                share|cite|improve this answer











                                                                                $endgroup$









                                                                                • 2




                                                                                  $begingroup$
                                                                                  If you are presenting this proof,write it nicely. $A=frac{n^2(n+1)^2}{4}$ and $B=frac{n(n+1)}{2}$,obviously $A=B^2$
                                                                                  $endgroup$
                                                                                  – Aang
                                                                                  Jun 27 '12 at 10:40










                                                                                • $begingroup$
                                                                                  @avatar: Dear Avatar; I will. Your proof is great and it looks analytically. Thanks. :-)
                                                                                  $endgroup$
                                                                                  – mrs
                                                                                  Jun 27 '12 at 10:48










                                                                                • $begingroup$
                                                                                  sorry, if you felt that i was being rude.
                                                                                  $endgroup$
                                                                                  – Aang
                                                                                  Jun 27 '12 at 10:53










                                                                                • $begingroup$
                                                                                  @avatar: Your proof lights mine. No Problem at all. WELCOME Avatar. :) :)
                                                                                  $endgroup$
                                                                                  – mrs
                                                                                  Jun 27 '12 at 10:57














                                                                                10












                                                                                10








                                                                                10





                                                                                $begingroup$

                                                                                We know that $$A=sum_1^n k^3=frac{1}{4}n^4+frac{1}{2}n^3+frac{1}{4}n^2$$ and $$B=sum_1^n k=frac{1}{2}n^2+frac{1}{2}n$$ $A-B^2=0$. :)






                                                                                share|cite|improve this answer











                                                                                $endgroup$



                                                                                We know that $$A=sum_1^n k^3=frac{1}{4}n^4+frac{1}{2}n^3+frac{1}{4}n^2$$ and $$B=sum_1^n k=frac{1}{2}n^2+frac{1}{2}n$$ $A-B^2=0$. :)







                                                                                share|cite|improve this answer














                                                                                share|cite|improve this answer



                                                                                share|cite|improve this answer








                                                                                edited Apr 1 '16 at 19:08









                                                                                Michael Hardy

                                                                                1




                                                                                1










                                                                                answered Jun 27 '12 at 10:30









                                                                                mrsmrs

                                                                                1




                                                                                1








                                                                                • 2




                                                                                  $begingroup$
                                                                                  If you are presenting this proof,write it nicely. $A=frac{n^2(n+1)^2}{4}$ and $B=frac{n(n+1)}{2}$,obviously $A=B^2$
                                                                                  $endgroup$
                                                                                  – Aang
                                                                                  Jun 27 '12 at 10:40










                                                                                • $begingroup$
                                                                                  @avatar: Dear Avatar; I will. Your proof is great and it looks analytically. Thanks. :-)
                                                                                  $endgroup$
                                                                                  – mrs
                                                                                  Jun 27 '12 at 10:48










                                                                                • $begingroup$
                                                                                  sorry, if you felt that i was being rude.
                                                                                  $endgroup$
                                                                                  – Aang
                                                                                  Jun 27 '12 at 10:53










                                                                                • $begingroup$
                                                                                  @avatar: Your proof lights mine. No Problem at all. WELCOME Avatar. :) :)
                                                                                  $endgroup$
                                                                                  – mrs
                                                                                  Jun 27 '12 at 10:57














                                                                                • 2




                                                                                  $begingroup$
                                                                                  If you are presenting this proof,write it nicely. $A=frac{n^2(n+1)^2}{4}$ and $B=frac{n(n+1)}{2}$,obviously $A=B^2$
                                                                                  $endgroup$
                                                                                  – Aang
                                                                                  Jun 27 '12 at 10:40










                                                                                • $begingroup$
                                                                                  @avatar: Dear Avatar; I will. Your proof is great and it looks analytically. Thanks. :-)
                                                                                  $endgroup$
                                                                                  – mrs
                                                                                  Jun 27 '12 at 10:48










                                                                                • $begingroup$
                                                                                  sorry, if you felt that i was being rude.
                                                                                  $endgroup$
                                                                                  – Aang
                                                                                  Jun 27 '12 at 10:53










                                                                                • $begingroup$
                                                                                  @avatar: Your proof lights mine. No Problem at all. WELCOME Avatar. :) :)
                                                                                  $endgroup$
                                                                                  – mrs
                                                                                  Jun 27 '12 at 10:57








                                                                                2




                                                                                2




                                                                                $begingroup$
                                                                                If you are presenting this proof,write it nicely. $A=frac{n^2(n+1)^2}{4}$ and $B=frac{n(n+1)}{2}$,obviously $A=B^2$
                                                                                $endgroup$
                                                                                – Aang
                                                                                Jun 27 '12 at 10:40




                                                                                $begingroup$
                                                                                If you are presenting this proof,write it nicely. $A=frac{n^2(n+1)^2}{4}$ and $B=frac{n(n+1)}{2}$,obviously $A=B^2$
                                                                                $endgroup$
                                                                                – Aang
                                                                                Jun 27 '12 at 10:40












                                                                                $begingroup$
                                                                                @avatar: Dear Avatar; I will. Your proof is great and it looks analytically. Thanks. :-)
                                                                                $endgroup$
                                                                                – mrs
                                                                                Jun 27 '12 at 10:48




                                                                                $begingroup$
                                                                                @avatar: Dear Avatar; I will. Your proof is great and it looks analytically. Thanks. :-)
                                                                                $endgroup$
                                                                                – mrs
                                                                                Jun 27 '12 at 10:48












                                                                                $begingroup$
                                                                                sorry, if you felt that i was being rude.
                                                                                $endgroup$
                                                                                – Aang
                                                                                Jun 27 '12 at 10:53




                                                                                $begingroup$
                                                                                sorry, if you felt that i was being rude.
                                                                                $endgroup$
                                                                                – Aang
                                                                                Jun 27 '12 at 10:53












                                                                                $begingroup$
                                                                                @avatar: Your proof lights mine. No Problem at all. WELCOME Avatar. :) :)
                                                                                $endgroup$
                                                                                – mrs
                                                                                Jun 27 '12 at 10:57




                                                                                $begingroup$
                                                                                @avatar: Your proof lights mine. No Problem at all. WELCOME Avatar. :) :)
                                                                                $endgroup$
                                                                                – mrs
                                                                                Jun 27 '12 at 10:57











                                                                                8












                                                                                $begingroup$

                                                                                $f(n)=1^3+2^3+3^3+cdots+n^3$



                                                                                $f(n-1)=1^3+2^3+3^3+cdots+(n-1)^3$



                                                                                $f(n)-f(n-1)=n^3$



                                                                                if $f(n)= (1+2+3+4+cdots+n)^2$ then



                                                                                $$f(n)-f(n-1)=(1+2+3+4+cdots+n)^2-(1+2+3+4+cdots+(n-1))^2$$



                                                                                using $a^2-b^2=(a+b)(a-b)$



                                                                                $f(n)-f(n-1)=$



                                                                                $=[1+1+2+2+3+3+4+4+cdots+(n-1)+(n-1)+n][1-1+2-2+3-3+4-4+cdots+(n-1)-(n-1)+n]=$



                                                                                $=[2(1+2+3+4+cdots+(n-1))+n]n=(2frac{n(n-1)}{2}+n)n=(n(n-1)+n)n=n^3$






                                                                                share|cite|improve this answer











                                                                                $endgroup$


















                                                                                  8












                                                                                  $begingroup$

                                                                                  $f(n)=1^3+2^3+3^3+cdots+n^3$



                                                                                  $f(n-1)=1^3+2^3+3^3+cdots+(n-1)^3$



                                                                                  $f(n)-f(n-1)=n^3$



                                                                                  if $f(n)= (1+2+3+4+cdots+n)^2$ then



                                                                                  $$f(n)-f(n-1)=(1+2+3+4+cdots+n)^2-(1+2+3+4+cdots+(n-1))^2$$



                                                                                  using $a^2-b^2=(a+b)(a-b)$



                                                                                  $f(n)-f(n-1)=$



                                                                                  $=[1+1+2+2+3+3+4+4+cdots+(n-1)+(n-1)+n][1-1+2-2+3-3+4-4+cdots+(n-1)-(n-1)+n]=$



                                                                                  $=[2(1+2+3+4+cdots+(n-1))+n]n=(2frac{n(n-1)}{2}+n)n=(n(n-1)+n)n=n^3$






                                                                                  share|cite|improve this answer











                                                                                  $endgroup$
















                                                                                    8












                                                                                    8








                                                                                    8





                                                                                    $begingroup$

                                                                                    $f(n)=1^3+2^3+3^3+cdots+n^3$



                                                                                    $f(n-1)=1^3+2^3+3^3+cdots+(n-1)^3$



                                                                                    $f(n)-f(n-1)=n^3$



                                                                                    if $f(n)= (1+2+3+4+cdots+n)^2$ then



                                                                                    $$f(n)-f(n-1)=(1+2+3+4+cdots+n)^2-(1+2+3+4+cdots+(n-1))^2$$



                                                                                    using $a^2-b^2=(a+b)(a-b)$



                                                                                    $f(n)-f(n-1)=$



                                                                                    $=[1+1+2+2+3+3+4+4+cdots+(n-1)+(n-1)+n][1-1+2-2+3-3+4-4+cdots+(n-1)-(n-1)+n]=$



                                                                                    $=[2(1+2+3+4+cdots+(n-1))+n]n=(2frac{n(n-1)}{2}+n)n=(n(n-1)+n)n=n^3$






                                                                                    share|cite|improve this answer











                                                                                    $endgroup$



                                                                                    $f(n)=1^3+2^3+3^3+cdots+n^3$



                                                                                    $f(n-1)=1^3+2^3+3^3+cdots+(n-1)^3$



                                                                                    $f(n)-f(n-1)=n^3$



                                                                                    if $f(n)= (1+2+3+4+cdots+n)^2$ then



                                                                                    $$f(n)-f(n-1)=(1+2+3+4+cdots+n)^2-(1+2+3+4+cdots+(n-1))^2$$



                                                                                    using $a^2-b^2=(a+b)(a-b)$



                                                                                    $f(n)-f(n-1)=$



                                                                                    $=[1+1+2+2+3+3+4+4+cdots+(n-1)+(n-1)+n][1-1+2-2+3-3+4-4+cdots+(n-1)-(n-1)+n]=$



                                                                                    $=[2(1+2+3+4+cdots+(n-1))+n]n=(2frac{n(n-1)}{2}+n)n=(n(n-1)+n)n=n^3$







                                                                                    share|cite|improve this answer














                                                                                    share|cite|improve this answer



                                                                                    share|cite|improve this answer








                                                                                    edited Jun 8 '16 at 22:56









                                                                                    Michael Hardy

                                                                                    1




                                                                                    1










                                                                                    answered Jun 27 '12 at 12:18









                                                                                    MathloverMathlover

                                                                                    6,25222469




                                                                                    6,25222469























                                                                                        6












                                                                                        $begingroup$

                                                                                        http://en.wikipedia.org/wiki/Faulhaber%27s_formula#Faulhaber_polynomials



                                                                                        If $p$ is odd, then $1^p+2^p+3^p+cdots+n^p$ is a polynomial function of $a=1+2+3+cdots+n$. If $p=3$, then then the sum is $a^2$; if $p=5$ then it's $(4a^3-a^2)/3$, and so on.






                                                                                        share|cite|improve this answer









                                                                                        $endgroup$













                                                                                        • $begingroup$
                                                                                          And $a$ is always a factor of $p$.
                                                                                          $endgroup$
                                                                                          – lhf
                                                                                          Sep 3 '11 at 13:26
















                                                                                        6












                                                                                        $begingroup$

                                                                                        http://en.wikipedia.org/wiki/Faulhaber%27s_formula#Faulhaber_polynomials



                                                                                        If $p$ is odd, then $1^p+2^p+3^p+cdots+n^p$ is a polynomial function of $a=1+2+3+cdots+n$. If $p=3$, then then the sum is $a^2$; if $p=5$ then it's $(4a^3-a^2)/3$, and so on.






                                                                                        share|cite|improve this answer









                                                                                        $endgroup$













                                                                                        • $begingroup$
                                                                                          And $a$ is always a factor of $p$.
                                                                                          $endgroup$
                                                                                          – lhf
                                                                                          Sep 3 '11 at 13:26














                                                                                        6












                                                                                        6








                                                                                        6





                                                                                        $begingroup$

                                                                                        http://en.wikipedia.org/wiki/Faulhaber%27s_formula#Faulhaber_polynomials



                                                                                        If $p$ is odd, then $1^p+2^p+3^p+cdots+n^p$ is a polynomial function of $a=1+2+3+cdots+n$. If $p=3$, then then the sum is $a^2$; if $p=5$ then it's $(4a^3-a^2)/3$, and so on.






                                                                                        share|cite|improve this answer









                                                                                        $endgroup$



                                                                                        http://en.wikipedia.org/wiki/Faulhaber%27s_formula#Faulhaber_polynomials



                                                                                        If $p$ is odd, then $1^p+2^p+3^p+cdots+n^p$ is a polynomial function of $a=1+2+3+cdots+n$. If $p=3$, then then the sum is $a^2$; if $p=5$ then it's $(4a^3-a^2)/3$, and so on.







                                                                                        share|cite|improve this answer












                                                                                        share|cite|improve this answer



                                                                                        share|cite|improve this answer










                                                                                        answered Sep 3 '11 at 1:29









                                                                                        Michael HardyMichael Hardy

                                                                                        1




                                                                                        1












                                                                                        • $begingroup$
                                                                                          And $a$ is always a factor of $p$.
                                                                                          $endgroup$
                                                                                          – lhf
                                                                                          Sep 3 '11 at 13:26


















                                                                                        • $begingroup$
                                                                                          And $a$ is always a factor of $p$.
                                                                                          $endgroup$
                                                                                          – lhf
                                                                                          Sep 3 '11 at 13:26
















                                                                                        $begingroup$
                                                                                        And $a$ is always a factor of $p$.
                                                                                        $endgroup$
                                                                                        – lhf
                                                                                        Sep 3 '11 at 13:26




                                                                                        $begingroup$
                                                                                        And $a$ is always a factor of $p$.
                                                                                        $endgroup$
                                                                                        – lhf
                                                                                        Sep 3 '11 at 13:26











                                                                                        5












                                                                                        $begingroup$

                                                                                        Chance would have it that I stumbled* upon this article today:



                                                                                        http://blogs.mathworks.com/loren/2010/03/04/nichomachuss-theorem/



                                                                                        It seems to answer your question.



                                                                                        (* That is, @AlgebraFact on Twitter posted a link)






                                                                                        share|cite|improve this answer









                                                                                        $endgroup$


















                                                                                          5












                                                                                          $begingroup$

                                                                                          Chance would have it that I stumbled* upon this article today:



                                                                                          http://blogs.mathworks.com/loren/2010/03/04/nichomachuss-theorem/



                                                                                          It seems to answer your question.



                                                                                          (* That is, @AlgebraFact on Twitter posted a link)






                                                                                          share|cite|improve this answer









                                                                                          $endgroup$
















                                                                                            5












                                                                                            5








                                                                                            5





                                                                                            $begingroup$

                                                                                            Chance would have it that I stumbled* upon this article today:



                                                                                            http://blogs.mathworks.com/loren/2010/03/04/nichomachuss-theorem/



                                                                                            It seems to answer your question.



                                                                                            (* That is, @AlgebraFact on Twitter posted a link)






                                                                                            share|cite|improve this answer









                                                                                            $endgroup$



                                                                                            Chance would have it that I stumbled* upon this article today:



                                                                                            http://blogs.mathworks.com/loren/2010/03/04/nichomachuss-theorem/



                                                                                            It seems to answer your question.



                                                                                            (* That is, @AlgebraFact on Twitter posted a link)







                                                                                            share|cite|improve this answer












                                                                                            share|cite|improve this answer



                                                                                            share|cite|improve this answer










                                                                                            answered Sep 2 '11 at 21:19









                                                                                            Fredrik MeyerFredrik Meyer

                                                                                            15.3k24165




                                                                                            15.3k24165























                                                                                                5












                                                                                                $begingroup$

                                                                                                The square in the identity is the area of the triangle below, while the cubes are the area's of the trapezoidal layers, with heights $k = 1, 2, cdots, n$
                                                                                                TriangleWithHeight 1+2+..+n



                                                                                                The trapezoids have area $k^3$ because their height equals $k$ and the $text{width}_{text{atHalfHeight}}$ consists of $k$ diagonals with width $k$:
                                                                                                Trapezoidal layer with height = k and width = k^2



                                                                                                The total of the triangle is its squared height $(1 + 2 + cdots + n)^2$, because this triangle can be turned into a square:
                                                                                                The triangle cut in two and recomposed as a square.



                                                                                                Therefore:
                                                                                                $(1 + 2 + cdots + n)^2 = sum_{k=1}^n k^3$ , $blacksquare$






                                                                                                share|cite|improve this answer











                                                                                                $endgroup$


















                                                                                                  5












                                                                                                  $begingroup$

                                                                                                  The square in the identity is the area of the triangle below, while the cubes are the area's of the trapezoidal layers, with heights $k = 1, 2, cdots, n$
                                                                                                  TriangleWithHeight 1+2+..+n



                                                                                                  The trapezoids have area $k^3$ because their height equals $k$ and the $text{width}_{text{atHalfHeight}}$ consists of $k$ diagonals with width $k$:
                                                                                                  Trapezoidal layer with height = k and width = k^2



                                                                                                  The total of the triangle is its squared height $(1 + 2 + cdots + n)^2$, because this triangle can be turned into a square:
                                                                                                  The triangle cut in two and recomposed as a square.



                                                                                                  Therefore:
                                                                                                  $(1 + 2 + cdots + n)^2 = sum_{k=1}^n k^3$ , $blacksquare$






                                                                                                  share|cite|improve this answer











                                                                                                  $endgroup$
















                                                                                                    5












                                                                                                    5








                                                                                                    5





                                                                                                    $begingroup$

                                                                                                    The square in the identity is the area of the triangle below, while the cubes are the area's of the trapezoidal layers, with heights $k = 1, 2, cdots, n$
                                                                                                    TriangleWithHeight 1+2+..+n



                                                                                                    The trapezoids have area $k^3$ because their height equals $k$ and the $text{width}_{text{atHalfHeight}}$ consists of $k$ diagonals with width $k$:
                                                                                                    Trapezoidal layer with height = k and width = k^2



                                                                                                    The total of the triangle is its squared height $(1 + 2 + cdots + n)^2$, because this triangle can be turned into a square:
                                                                                                    The triangle cut in two and recomposed as a square.



                                                                                                    Therefore:
                                                                                                    $(1 + 2 + cdots + n)^2 = sum_{k=1}^n k^3$ , $blacksquare$






                                                                                                    share|cite|improve this answer











                                                                                                    $endgroup$



                                                                                                    The square in the identity is the area of the triangle below, while the cubes are the area's of the trapezoidal layers, with heights $k = 1, 2, cdots, n$
                                                                                                    TriangleWithHeight 1+2+..+n



                                                                                                    The trapezoids have area $k^3$ because their height equals $k$ and the $text{width}_{text{atHalfHeight}}$ consists of $k$ diagonals with width $k$:
                                                                                                    Trapezoidal layer with height = k and width = k^2



                                                                                                    The total of the triangle is its squared height $(1 + 2 + cdots + n)^2$, because this triangle can be turned into a square:
                                                                                                    The triangle cut in two and recomposed as a square.



                                                                                                    Therefore:
                                                                                                    $(1 + 2 + cdots + n)^2 = sum_{k=1}^n k^3$ , $blacksquare$







                                                                                                    share|cite|improve this answer














                                                                                                    share|cite|improve this answer



                                                                                                    share|cite|improve this answer








                                                                                                    edited Dec 18 '15 at 15:52

























                                                                                                    answered Dec 18 '15 at 1:38









                                                                                                    Job BouwmanJob Bouwman

                                                                                                    966811




                                                                                                    966811























                                                                                                        5












                                                                                                        $begingroup$

                                                                                                        One way to show that
                                                                                                        $$sum_{i=1}^n i^3 = bigg(sum_{i=1}^n i bigg)^2$$
                                                                                                        is to add up the entries in the multiplication tables, but first we need to show that
                                                                                                        $$1+2+3+dots+n+dots+3+2+1 = n^2$$
                                                                                                        For this, see the image below (n=7)enter image description here
                                                                                                        $$7^2=color{green}{1+2+3+4+5+6+7}color{red}{+6+5+4+3+2+1}$$
                                                                                                        Next, consider the standard multiplication table that we are all familiar with.The graphic shows the table up to the 9s.



                                                                                                        enter image description here



                                                                                                        We can add up the entries in any order that we wish.
                                                                                                        One way would be to add up a series of Ls (the 6th L ($L_6$) is highlighted in yellow).
                                                                                                        $$begin{align}
                                                                                                        L_6 &= 6+12+18+24+30+36+30+24+18+12+6\
                                                                                                        &=6(1+2+3+4+5+6+5+4+3+2+1)\
                                                                                                        &=6(6^2)\
                                                                                                        &=6^3
                                                                                                        end{align}$$
                                                                                                        And the sum of all the entries in the table becomes
                                                                                                        $$sum_{i=1}^n L_i = sum_{i=1}^n i^3$$
                                                                                                        Alternatively, we could just add up each row. The 6th row (R_6) would be
                                                                                                        $$begin{align}
                                                                                                        R_6 &= 6+12+18+24+30+36+42+48+54\
                                                                                                        &= 6(1+2+3+4+5+6+7+8+9)\
                                                                                                        &= 6sum_{i=1}^9 i
                                                                                                        end{align}$$
                                                                                                        And the sum of all the entries becomes
                                                                                                        $$sum_{i=1}^n R_i = sum_{i=1}^n bigg(isum_{j=1}^n jbigg)=bigg(sum_{j=1}^n jbigg)bigg(sum_{i=1}^n ibigg)=bigg(sum_{i=1}^n ibigg)^2$$
                                                                                                        Thus we have
                                                                                                        $$sum_{i=1}^n i^3 = sum_{i=1}^n L_i = sum_{i=1}^n R_i=bigg(sum_{i=1}^n ibigg)^2$$






                                                                                                        share|cite|improve this answer











                                                                                                        $endgroup$


















                                                                                                          5












                                                                                                          $begingroup$

                                                                                                          One way to show that
                                                                                                          $$sum_{i=1}^n i^3 = bigg(sum_{i=1}^n i bigg)^2$$
                                                                                                          is to add up the entries in the multiplication tables, but first we need to show that
                                                                                                          $$1+2+3+dots+n+dots+3+2+1 = n^2$$
                                                                                                          For this, see the image below (n=7)enter image description here
                                                                                                          $$7^2=color{green}{1+2+3+4+5+6+7}color{red}{+6+5+4+3+2+1}$$
                                                                                                          Next, consider the standard multiplication table that we are all familiar with.The graphic shows the table up to the 9s.



                                                                                                          enter image description here



                                                                                                          We can add up the entries in any order that we wish.
                                                                                                          One way would be to add up a series of Ls (the 6th L ($L_6$) is highlighted in yellow).
                                                                                                          $$begin{align}
                                                                                                          L_6 &= 6+12+18+24+30+36+30+24+18+12+6\
                                                                                                          &=6(1+2+3+4+5+6+5+4+3+2+1)\
                                                                                                          &=6(6^2)\
                                                                                                          &=6^3
                                                                                                          end{align}$$
                                                                                                          And the sum of all the entries in the table becomes
                                                                                                          $$sum_{i=1}^n L_i = sum_{i=1}^n i^3$$
                                                                                                          Alternatively, we could just add up each row. The 6th row (R_6) would be
                                                                                                          $$begin{align}
                                                                                                          R_6 &= 6+12+18+24+30+36+42+48+54\
                                                                                                          &= 6(1+2+3+4+5+6+7+8+9)\
                                                                                                          &= 6sum_{i=1}^9 i
                                                                                                          end{align}$$
                                                                                                          And the sum of all the entries becomes
                                                                                                          $$sum_{i=1}^n R_i = sum_{i=1}^n bigg(isum_{j=1}^n jbigg)=bigg(sum_{j=1}^n jbigg)bigg(sum_{i=1}^n ibigg)=bigg(sum_{i=1}^n ibigg)^2$$
                                                                                                          Thus we have
                                                                                                          $$sum_{i=1}^n i^3 = sum_{i=1}^n L_i = sum_{i=1}^n R_i=bigg(sum_{i=1}^n ibigg)^2$$






                                                                                                          share|cite|improve this answer











                                                                                                          $endgroup$
















                                                                                                            5












                                                                                                            5








                                                                                                            5





                                                                                                            $begingroup$

                                                                                                            One way to show that
                                                                                                            $$sum_{i=1}^n i^3 = bigg(sum_{i=1}^n i bigg)^2$$
                                                                                                            is to add up the entries in the multiplication tables, but first we need to show that
                                                                                                            $$1+2+3+dots+n+dots+3+2+1 = n^2$$
                                                                                                            For this, see the image below (n=7)enter image description here
                                                                                                            $$7^2=color{green}{1+2+3+4+5+6+7}color{red}{+6+5+4+3+2+1}$$
                                                                                                            Next, consider the standard multiplication table that we are all familiar with.The graphic shows the table up to the 9s.



                                                                                                            enter image description here



                                                                                                            We can add up the entries in any order that we wish.
                                                                                                            One way would be to add up a series of Ls (the 6th L ($L_6$) is highlighted in yellow).
                                                                                                            $$begin{align}
                                                                                                            L_6 &= 6+12+18+24+30+36+30+24+18+12+6\
                                                                                                            &=6(1+2+3+4+5+6+5+4+3+2+1)\
                                                                                                            &=6(6^2)\
                                                                                                            &=6^3
                                                                                                            end{align}$$
                                                                                                            And the sum of all the entries in the table becomes
                                                                                                            $$sum_{i=1}^n L_i = sum_{i=1}^n i^3$$
                                                                                                            Alternatively, we could just add up each row. The 6th row (R_6) would be
                                                                                                            $$begin{align}
                                                                                                            R_6 &= 6+12+18+24+30+36+42+48+54\
                                                                                                            &= 6(1+2+3+4+5+6+7+8+9)\
                                                                                                            &= 6sum_{i=1}^9 i
                                                                                                            end{align}$$
                                                                                                            And the sum of all the entries becomes
                                                                                                            $$sum_{i=1}^n R_i = sum_{i=1}^n bigg(isum_{j=1}^n jbigg)=bigg(sum_{j=1}^n jbigg)bigg(sum_{i=1}^n ibigg)=bigg(sum_{i=1}^n ibigg)^2$$
                                                                                                            Thus we have
                                                                                                            $$sum_{i=1}^n i^3 = sum_{i=1}^n L_i = sum_{i=1}^n R_i=bigg(sum_{i=1}^n ibigg)^2$$






                                                                                                            share|cite|improve this answer











                                                                                                            $endgroup$



                                                                                                            One way to show that
                                                                                                            $$sum_{i=1}^n i^3 = bigg(sum_{i=1}^n i bigg)^2$$
                                                                                                            is to add up the entries in the multiplication tables, but first we need to show that
                                                                                                            $$1+2+3+dots+n+dots+3+2+1 = n^2$$
                                                                                                            For this, see the image below (n=7)enter image description here
                                                                                                            $$7^2=color{green}{1+2+3+4+5+6+7}color{red}{+6+5+4+3+2+1}$$
                                                                                                            Next, consider the standard multiplication table that we are all familiar with.The graphic shows the table up to the 9s.



                                                                                                            enter image description here



                                                                                                            We can add up the entries in any order that we wish.
                                                                                                            One way would be to add up a series of Ls (the 6th L ($L_6$) is highlighted in yellow).
                                                                                                            $$begin{align}
                                                                                                            L_6 &= 6+12+18+24+30+36+30+24+18+12+6\
                                                                                                            &=6(1+2+3+4+5+6+5+4+3+2+1)\
                                                                                                            &=6(6^2)\
                                                                                                            &=6^3
                                                                                                            end{align}$$
                                                                                                            And the sum of all the entries in the table becomes
                                                                                                            $$sum_{i=1}^n L_i = sum_{i=1}^n i^3$$
                                                                                                            Alternatively, we could just add up each row. The 6th row (R_6) would be
                                                                                                            $$begin{align}
                                                                                                            R_6 &= 6+12+18+24+30+36+42+48+54\
                                                                                                            &= 6(1+2+3+4+5+6+7+8+9)\
                                                                                                            &= 6sum_{i=1}^9 i
                                                                                                            end{align}$$
                                                                                                            And the sum of all the entries becomes
                                                                                                            $$sum_{i=1}^n R_i = sum_{i=1}^n bigg(isum_{j=1}^n jbigg)=bigg(sum_{j=1}^n jbigg)bigg(sum_{i=1}^n ibigg)=bigg(sum_{i=1}^n ibigg)^2$$
                                                                                                            Thus we have
                                                                                                            $$sum_{i=1}^n i^3 = sum_{i=1}^n L_i = sum_{i=1}^n R_i=bigg(sum_{i=1}^n ibigg)^2$$







                                                                                                            share|cite|improve this answer














                                                                                                            share|cite|improve this answer



                                                                                                            share|cite|improve this answer








                                                                                                            edited Jun 8 '16 at 22:58









                                                                                                            Michael Hardy

                                                                                                            1




                                                                                                            1










                                                                                                            answered Dec 27 '14 at 6:27









                                                                                                            John JoyJohn Joy

                                                                                                            6,29911727




                                                                                                            6,29911727























                                                                                                                4












                                                                                                                $begingroup$

                                                                                                                square triangular proof without words



                                                                                                                This is about the same proof as here, the presentation is a bit different though. This is another way to make $k^3$ appear than what was shown here, here and here.






                                                                                                                share|cite|improve this answer











                                                                                                                $endgroup$


















                                                                                                                  4












                                                                                                                  $begingroup$

                                                                                                                  square triangular proof without words



                                                                                                                  This is about the same proof as here, the presentation is a bit different though. This is another way to make $k^3$ appear than what was shown here, here and here.






                                                                                                                  share|cite|improve this answer











                                                                                                                  $endgroup$
















                                                                                                                    4












                                                                                                                    4








                                                                                                                    4





                                                                                                                    $begingroup$

                                                                                                                    square triangular proof without words



                                                                                                                    This is about the same proof as here, the presentation is a bit different though. This is another way to make $k^3$ appear than what was shown here, here and here.






                                                                                                                    share|cite|improve this answer











                                                                                                                    $endgroup$



                                                                                                                    square triangular proof without words



                                                                                                                    This is about the same proof as here, the presentation is a bit different though. This is another way to make $k^3$ appear than what was shown here, here and here.







                                                                                                                    share|cite|improve this answer














                                                                                                                    share|cite|improve this answer



                                                                                                                    share|cite|improve this answer








                                                                                                                    edited Apr 13 '17 at 12:20









                                                                                                                    Community

                                                                                                                    1




                                                                                                                    1










                                                                                                                    answered Sep 4 '11 at 8:12









                                                                                                                    WokWok

                                                                                                                    1,5781323




                                                                                                                    1,5781323























                                                                                                                        4












                                                                                                                        $begingroup$

                                                                                                                        Here's a simple bijective proof of a different sort:



                                                                                                                        Consider a staircase with $n$ steps, built out of $sum_{k=1}^n k$ blocks. In other words, take the set ${(i,j) in mathbb{Z}timesmathbb{Z}: i + j leq n, i > 0, j > 0}$.



                                                                                                                        Then $left(sum_{k=1}^n kright)^2$ is the number of ordered pairs $(B_1,B_2)$ of blocks.



                                                                                                                        And $sum_{k=1}^n k^3$ is the number of ordered $4$-tuples $(k,b_1,b_2,b_3)$, where $k in {1,ldots,n}$, and $b_1$ is along the $k$-th diagonal $b_1 in {(k+1-j,j): j in {1,ldots,k}}$, and $b_2$ is along the bottom $b_2 in {(j,1): j in {1,ldots, k}}$ and $b_3$ is along the left side $b_3 in {(1,j): j in {1, ldots, k}}$.



                                                                                                                        The bijection:



                                                                                                                        Given an ordered tuple $(k,b_1,b_2,b_3)$, let $A_1 = b_1$ and let $A_2$ given by $b_2$ and $b_3$ as its $x$ and $y$ coordinates, so if $b_2 = (i,1)$ and $b_3 = (1,j)$ then $A_2 = (i,j)$.



                                                                                                                        Case 1: $A_2$ is on or below the $k$-th diagonal. Then let $(B_1, B_2) = (A_1, A_2)$.



                                                                                                                        Case 2: $A_2$ is above the $k$-the diagonal. Then let $A_2'$ be the reflection across the $k$-th diagonal of $A_2$. That is, if $A_2 = (i,j)$ then $A_2' = (k+1-j,k+1-i)$. Then let $(B_1, B_2) = (A_2', A_1)$.



                                                                                                                        The inverse:



                                                                                                                        To get the inverse, take whichever of $B_1$ and $B_2$ is on a higher diagonal (i.e. has greater sum of its coordinates), taking $B_1$ in case of ties, and let that be $b_1$ and let $k$ be the sum of the coordinates of $b_1$.



                                                                                                                        Case 1: If $B_1$ is used: Take $B_2$ and let $b_2$ and $b_3$ be given by points with the same the $x$- and $y$-coordinates, respectively, as $B_2$.



                                                                                                                        Case 2: If $B_2$ is used: Take $B_1'$ (i.e. the reflection across the $k$-th diagonal, as above) and let $b_2$ and $b_3$ be given by points with the same the $x$- and $y$-coordinates, respectively, as $B_1'$.






                                                                                                                        share|cite|improve this answer











                                                                                                                        $endgroup$


















                                                                                                                          4












                                                                                                                          $begingroup$

                                                                                                                          Here's a simple bijective proof of a different sort:



                                                                                                                          Consider a staircase with $n$ steps, built out of $sum_{k=1}^n k$ blocks. In other words, take the set ${(i,j) in mathbb{Z}timesmathbb{Z}: i + j leq n, i > 0, j > 0}$.



                                                                                                                          Then $left(sum_{k=1}^n kright)^2$ is the number of ordered pairs $(B_1,B_2)$ of blocks.



                                                                                                                          And $sum_{k=1}^n k^3$ is the number of ordered $4$-tuples $(k,b_1,b_2,b_3)$, where $k in {1,ldots,n}$, and $b_1$ is along the $k$-th diagonal $b_1 in {(k+1-j,j): j in {1,ldots,k}}$, and $b_2$ is along the bottom $b_2 in {(j,1): j in {1,ldots, k}}$ and $b_3$ is along the left side $b_3 in {(1,j): j in {1, ldots, k}}$.



                                                                                                                          The bijection:



                                                                                                                          Given an ordered tuple $(k,b_1,b_2,b_3)$, let $A_1 = b_1$ and let $A_2$ given by $b_2$ and $b_3$ as its $x$ and $y$ coordinates, so if $b_2 = (i,1)$ and $b_3 = (1,j)$ then $A_2 = (i,j)$.



                                                                                                                          Case 1: $A_2$ is on or below the $k$-th diagonal. Then let $(B_1, B_2) = (A_1, A_2)$.



                                                                                                                          Case 2: $A_2$ is above the $k$-the diagonal. Then let $A_2'$ be the reflection across the $k$-th diagonal of $A_2$. That is, if $A_2 = (i,j)$ then $A_2' = (k+1-j,k+1-i)$. Then let $(B_1, B_2) = (A_2', A_1)$.



                                                                                                                          The inverse:



                                                                                                                          To get the inverse, take whichever of $B_1$ and $B_2$ is on a higher diagonal (i.e. has greater sum of its coordinates), taking $B_1$ in case of ties, and let that be $b_1$ and let $k$ be the sum of the coordinates of $b_1$.



                                                                                                                          Case 1: If $B_1$ is used: Take $B_2$ and let $b_2$ and $b_3$ be given by points with the same the $x$- and $y$-coordinates, respectively, as $B_2$.



                                                                                                                          Case 2: If $B_2$ is used: Take $B_1'$ (i.e. the reflection across the $k$-th diagonal, as above) and let $b_2$ and $b_3$ be given by points with the same the $x$- and $y$-coordinates, respectively, as $B_1'$.






                                                                                                                          share|cite|improve this answer











                                                                                                                          $endgroup$
















                                                                                                                            4












                                                                                                                            4








                                                                                                                            4





                                                                                                                            $begingroup$

                                                                                                                            Here's a simple bijective proof of a different sort:



                                                                                                                            Consider a staircase with $n$ steps, built out of $sum_{k=1}^n k$ blocks. In other words, take the set ${(i,j) in mathbb{Z}timesmathbb{Z}: i + j leq n, i > 0, j > 0}$.



                                                                                                                            Then $left(sum_{k=1}^n kright)^2$ is the number of ordered pairs $(B_1,B_2)$ of blocks.



                                                                                                                            And $sum_{k=1}^n k^3$ is the number of ordered $4$-tuples $(k,b_1,b_2,b_3)$, where $k in {1,ldots,n}$, and $b_1$ is along the $k$-th diagonal $b_1 in {(k+1-j,j): j in {1,ldots,k}}$, and $b_2$ is along the bottom $b_2 in {(j,1): j in {1,ldots, k}}$ and $b_3$ is along the left side $b_3 in {(1,j): j in {1, ldots, k}}$.



                                                                                                                            The bijection:



                                                                                                                            Given an ordered tuple $(k,b_1,b_2,b_3)$, let $A_1 = b_1$ and let $A_2$ given by $b_2$ and $b_3$ as its $x$ and $y$ coordinates, so if $b_2 = (i,1)$ and $b_3 = (1,j)$ then $A_2 = (i,j)$.



                                                                                                                            Case 1: $A_2$ is on or below the $k$-th diagonal. Then let $(B_1, B_2) = (A_1, A_2)$.



                                                                                                                            Case 2: $A_2$ is above the $k$-the diagonal. Then let $A_2'$ be the reflection across the $k$-th diagonal of $A_2$. That is, if $A_2 = (i,j)$ then $A_2' = (k+1-j,k+1-i)$. Then let $(B_1, B_2) = (A_2', A_1)$.



                                                                                                                            The inverse:



                                                                                                                            To get the inverse, take whichever of $B_1$ and $B_2$ is on a higher diagonal (i.e. has greater sum of its coordinates), taking $B_1$ in case of ties, and let that be $b_1$ and let $k$ be the sum of the coordinates of $b_1$.



                                                                                                                            Case 1: If $B_1$ is used: Take $B_2$ and let $b_2$ and $b_3$ be given by points with the same the $x$- and $y$-coordinates, respectively, as $B_2$.



                                                                                                                            Case 2: If $B_2$ is used: Take $B_1'$ (i.e. the reflection across the $k$-th diagonal, as above) and let $b_2$ and $b_3$ be given by points with the same the $x$- and $y$-coordinates, respectively, as $B_1'$.






                                                                                                                            share|cite|improve this answer











                                                                                                                            $endgroup$



                                                                                                                            Here's a simple bijective proof of a different sort:



                                                                                                                            Consider a staircase with $n$ steps, built out of $sum_{k=1}^n k$ blocks. In other words, take the set ${(i,j) in mathbb{Z}timesmathbb{Z}: i + j leq n, i > 0, j > 0}$.



                                                                                                                            Then $left(sum_{k=1}^n kright)^2$ is the number of ordered pairs $(B_1,B_2)$ of blocks.



                                                                                                                            And $sum_{k=1}^n k^3$ is the number of ordered $4$-tuples $(k,b_1,b_2,b_3)$, where $k in {1,ldots,n}$, and $b_1$ is along the $k$-th diagonal $b_1 in {(k+1-j,j): j in {1,ldots,k}}$, and $b_2$ is along the bottom $b_2 in {(j,1): j in {1,ldots, k}}$ and $b_3$ is along the left side $b_3 in {(1,j): j in {1, ldots, k}}$.



                                                                                                                            The bijection:



                                                                                                                            Given an ordered tuple $(k,b_1,b_2,b_3)$, let $A_1 = b_1$ and let $A_2$ given by $b_2$ and $b_3$ as its $x$ and $y$ coordinates, so if $b_2 = (i,1)$ and $b_3 = (1,j)$ then $A_2 = (i,j)$.



                                                                                                                            Case 1: $A_2$ is on or below the $k$-th diagonal. Then let $(B_1, B_2) = (A_1, A_2)$.



                                                                                                                            Case 2: $A_2$ is above the $k$-the diagonal. Then let $A_2'$ be the reflection across the $k$-th diagonal of $A_2$. That is, if $A_2 = (i,j)$ then $A_2' = (k+1-j,k+1-i)$. Then let $(B_1, B_2) = (A_2', A_1)$.



                                                                                                                            The inverse:



                                                                                                                            To get the inverse, take whichever of $B_1$ and $B_2$ is on a higher diagonal (i.e. has greater sum of its coordinates), taking $B_1$ in case of ties, and let that be $b_1$ and let $k$ be the sum of the coordinates of $b_1$.



                                                                                                                            Case 1: If $B_1$ is used: Take $B_2$ and let $b_2$ and $b_3$ be given by points with the same the $x$- and $y$-coordinates, respectively, as $B_2$.



                                                                                                                            Case 2: If $B_2$ is used: Take $B_1'$ (i.e. the reflection across the $k$-th diagonal, as above) and let $b_2$ and $b_3$ be given by points with the same the $x$- and $y$-coordinates, respectively, as $B_1'$.







                                                                                                                            share|cite|improve this answer














                                                                                                                            share|cite|improve this answer



                                                                                                                            share|cite|improve this answer








                                                                                                                            edited Mar 23 '15 at 2:18

























                                                                                                                            answered Mar 23 '15 at 2:13









                                                                                                                            aesaes

                                                                                                                            6,27911634




                                                                                                                            6,27911634























                                                                                                                                0












                                                                                                                                $begingroup$

                                                                                                                                After many years I still think the best way to solve this kind of problem in a natural and systematic way is to view it as a recurrence relation with constants coefficients, in this case, $x_n = x_{n-1}+n^3$. The way I learnt to do so was by using characteristic polynomial but you may prefer some other method...






                                                                                                                                share|cite|improve this answer











                                                                                                                                $endgroup$


















                                                                                                                                  0












                                                                                                                                  $begingroup$

                                                                                                                                  After many years I still think the best way to solve this kind of problem in a natural and systematic way is to view it as a recurrence relation with constants coefficients, in this case, $x_n = x_{n-1}+n^3$. The way I learnt to do so was by using characteristic polynomial but you may prefer some other method...






                                                                                                                                  share|cite|improve this answer











                                                                                                                                  $endgroup$
















                                                                                                                                    0












                                                                                                                                    0








                                                                                                                                    0





                                                                                                                                    $begingroup$

                                                                                                                                    After many years I still think the best way to solve this kind of problem in a natural and systematic way is to view it as a recurrence relation with constants coefficients, in this case, $x_n = x_{n-1}+n^3$. The way I learnt to do so was by using characteristic polynomial but you may prefer some other method...






                                                                                                                                    share|cite|improve this answer











                                                                                                                                    $endgroup$



                                                                                                                                    After many years I still think the best way to solve this kind of problem in a natural and systematic way is to view it as a recurrence relation with constants coefficients, in this case, $x_n = x_{n-1}+n^3$. The way I learnt to do so was by using characteristic polynomial but you may prefer some other method...







                                                                                                                                    share|cite|improve this answer














                                                                                                                                    share|cite|improve this answer



                                                                                                                                    share|cite|improve this answer








                                                                                                                                    edited Mar 29 '18 at 14:23

























                                                                                                                                    answered Mar 12 '18 at 0:24









                                                                                                                                    JavierJavier

                                                                                                                                    2,07521234




                                                                                                                                    2,07521234























                                                                                                                                        0












                                                                                                                                        $begingroup$

                                                                                                                                        For every $kinmathbb{N}$
                                                                                                                                        $$(k+1)^4=k^4+4k^3++6k^2+4k+1$$
                                                                                                                                        therefore
                                                                                                                                        $$sum_{k=1}^n(k+1)^4=sum_{k=1}^nk^4+4sum_{k=1}^nk^3+6sum_{k=1}^nk^2+4sum_{k=1}^nk+sum_{k=1}^n1$$
                                                                                                                                        which is equivalent to
                                                                                                                                        $$sum_{k=1}^nk^4+(n+1)^4-1=sum_{k=1}^nk^4+4sum_{k=1}^nk^3+6sum_{k=1}^nk^2+2n(n+1)+n$$
                                                                                                                                        After simplifications we obtain
                                                                                                                                        $$4sum_{k=1}^nk^3=(n+1)^4-1-2n(n+1)-n-6sum_{k=1}^nk^2=n^4+4n^3+4n^2+n-6sum_{k=1}^nk^2$$
                                                                                                                                        Using
                                                                                                                                        $$sum_{k=1}^nk^2=frac{n(n+1)(2n+1)}{6}hspace{0.2cm}text{and}hspace{0.2cm}sum_{k=1}^nk=frac{n(n+1)}{2}$$
                                                                                                                                        we get
                                                                                                                                        $$4sum_{k=1}^nk^3=n^4+4n^3+4n^2+n-6sum_{k=1}^nk^2\=n^4+4n^3+4n^2+n-n(n+1)(2n+1)\=n^4+2n^3+n^2=n^2(n+1)^2$$
                                                                                                                                        Finally
                                                                                                                                        $$sum_{k=1}^nk^3=frac{n^2(n+1)^2}{4}=Big(frac{n(n+1)}{2}Big)^2=Big(sum_{k=1}^nkBig)^2$$






                                                                                                                                        share|cite|improve this answer









                                                                                                                                        $endgroup$


















                                                                                                                                          0












                                                                                                                                          $begingroup$

                                                                                                                                          For every $kinmathbb{N}$
                                                                                                                                          $$(k+1)^4=k^4+4k^3++6k^2+4k+1$$
                                                                                                                                          therefore
                                                                                                                                          $$sum_{k=1}^n(k+1)^4=sum_{k=1}^nk^4+4sum_{k=1}^nk^3+6sum_{k=1}^nk^2+4sum_{k=1}^nk+sum_{k=1}^n1$$
                                                                                                                                          which is equivalent to
                                                                                                                                          $$sum_{k=1}^nk^4+(n+1)^4-1=sum_{k=1}^nk^4+4sum_{k=1}^nk^3+6sum_{k=1}^nk^2+2n(n+1)+n$$
                                                                                                                                          After simplifications we obtain
                                                                                                                                          $$4sum_{k=1}^nk^3=(n+1)^4-1-2n(n+1)-n-6sum_{k=1}^nk^2=n^4+4n^3+4n^2+n-6sum_{k=1}^nk^2$$
                                                                                                                                          Using
                                                                                                                                          $$sum_{k=1}^nk^2=frac{n(n+1)(2n+1)}{6}hspace{0.2cm}text{and}hspace{0.2cm}sum_{k=1}^nk=frac{n(n+1)}{2}$$
                                                                                                                                          we get
                                                                                                                                          $$4sum_{k=1}^nk^3=n^4+4n^3+4n^2+n-6sum_{k=1}^nk^2\=n^4+4n^3+4n^2+n-n(n+1)(2n+1)\=n^4+2n^3+n^2=n^2(n+1)^2$$
                                                                                                                                          Finally
                                                                                                                                          $$sum_{k=1}^nk^3=frac{n^2(n+1)^2}{4}=Big(frac{n(n+1)}{2}Big)^2=Big(sum_{k=1}^nkBig)^2$$






                                                                                                                                          share|cite|improve this answer









                                                                                                                                          $endgroup$
















                                                                                                                                            0












                                                                                                                                            0








                                                                                                                                            0





                                                                                                                                            $begingroup$

                                                                                                                                            For every $kinmathbb{N}$
                                                                                                                                            $$(k+1)^4=k^4+4k^3++6k^2+4k+1$$
                                                                                                                                            therefore
                                                                                                                                            $$sum_{k=1}^n(k+1)^4=sum_{k=1}^nk^4+4sum_{k=1}^nk^3+6sum_{k=1}^nk^2+4sum_{k=1}^nk+sum_{k=1}^n1$$
                                                                                                                                            which is equivalent to
                                                                                                                                            $$sum_{k=1}^nk^4+(n+1)^4-1=sum_{k=1}^nk^4+4sum_{k=1}^nk^3+6sum_{k=1}^nk^2+2n(n+1)+n$$
                                                                                                                                            After simplifications we obtain
                                                                                                                                            $$4sum_{k=1}^nk^3=(n+1)^4-1-2n(n+1)-n-6sum_{k=1}^nk^2=n^4+4n^3+4n^2+n-6sum_{k=1}^nk^2$$
                                                                                                                                            Using
                                                                                                                                            $$sum_{k=1}^nk^2=frac{n(n+1)(2n+1)}{6}hspace{0.2cm}text{and}hspace{0.2cm}sum_{k=1}^nk=frac{n(n+1)}{2}$$
                                                                                                                                            we get
                                                                                                                                            $$4sum_{k=1}^nk^3=n^4+4n^3+4n^2+n-6sum_{k=1}^nk^2\=n^4+4n^3+4n^2+n-n(n+1)(2n+1)\=n^4+2n^3+n^2=n^2(n+1)^2$$
                                                                                                                                            Finally
                                                                                                                                            $$sum_{k=1}^nk^3=frac{n^2(n+1)^2}{4}=Big(frac{n(n+1)}{2}Big)^2=Big(sum_{k=1}^nkBig)^2$$






                                                                                                                                            share|cite|improve this answer









                                                                                                                                            $endgroup$



                                                                                                                                            For every $kinmathbb{N}$
                                                                                                                                            $$(k+1)^4=k^4+4k^3++6k^2+4k+1$$
                                                                                                                                            therefore
                                                                                                                                            $$sum_{k=1}^n(k+1)^4=sum_{k=1}^nk^4+4sum_{k=1}^nk^3+6sum_{k=1}^nk^2+4sum_{k=1}^nk+sum_{k=1}^n1$$
                                                                                                                                            which is equivalent to
                                                                                                                                            $$sum_{k=1}^nk^4+(n+1)^4-1=sum_{k=1}^nk^4+4sum_{k=1}^nk^3+6sum_{k=1}^nk^2+2n(n+1)+n$$
                                                                                                                                            After simplifications we obtain
                                                                                                                                            $$4sum_{k=1}^nk^3=(n+1)^4-1-2n(n+1)-n-6sum_{k=1}^nk^2=n^4+4n^3+4n^2+n-6sum_{k=1}^nk^2$$
                                                                                                                                            Using
                                                                                                                                            $$sum_{k=1}^nk^2=frac{n(n+1)(2n+1)}{6}hspace{0.2cm}text{and}hspace{0.2cm}sum_{k=1}^nk=frac{n(n+1)}{2}$$
                                                                                                                                            we get
                                                                                                                                            $$4sum_{k=1}^nk^3=n^4+4n^3+4n^2+n-6sum_{k=1}^nk^2\=n^4+4n^3+4n^2+n-n(n+1)(2n+1)\=n^4+2n^3+n^2=n^2(n+1)^2$$
                                                                                                                                            Finally
                                                                                                                                            $$sum_{k=1}^nk^3=frac{n^2(n+1)^2}{4}=Big(frac{n(n+1)}{2}Big)^2=Big(sum_{k=1}^nkBig)^2$$







                                                                                                                                            share|cite|improve this answer












                                                                                                                                            share|cite|improve this answer



                                                                                                                                            share|cite|improve this answer










                                                                                                                                            answered Mar 29 '18 at 14:50









                                                                                                                                            ArianArian

                                                                                                                                            5,320917




                                                                                                                                            5,320917























                                                                                                                                                0












                                                                                                                                                $begingroup$

                                                                                                                                                We begin by writing $k^3$ in a more clever fashion: $k^3 = k(k-1)(k-2) + 3k^2 - 2k$ :



                                                                                                                                                $$sum_{k=0}^n k^3 = sum_{k=0}^n k(k-1)(k-2) + 3k^2 -2k$$



                                                                                                                                                Distributing the summation we obtain: $$ sum_{k=0}^n k(k-1)(k-2) + sum_{k=0}^n 3k^2 - sum_{k=0}^n 2k$$



                                                                                                                                                Notice, $$k(k-1)(k-2) = frac {k!}{(k-3)!}$$
                                                                                                                                                Now we have
                                                                                                                                                $$sum_{k=0}^n frac {k!}{(k-3)!} + sum_{k=0}^n 3k^2 - sum_{k=0}^n 2k$$
                                                                                                                                                Notice that we have nearly obtained the binomial expansion of K choose 3, all we need to do is divide by 3! So we offset this by also taking the product of 3!
                                                                                                                                                $$sum_{k=0}^n frac {k!}{(k-3)!} = 3!sum_{k=0}^n frac {k!}{(k-3)!3!} = 3!sum_{k=0}^nbinom{k}{3} = 3!binom{n+1}{4}$$
                                                                                                                                                We have now obtained
                                                                                                                                                $$sum_{k=0}^n k^3 = 3!binom{n+1}{4} + 3sum_{k=0}^n k^2 - 2sum_{k=0}^n k$$
                                                                                                                                                Focusing solely on the right-hand side we have
                                                                                                                                                $$6biggl(frac {(n+1)!}{(n-3)!4!}biggr) + 3sum_{k=0}^n k^2 - 2sum_{k=0}^n k$$
                                                                                                                                                Assuming we already know the sum of the sequence of integers and squared integers (the 2 sums we have left) we have
                                                                                                                                                $$ frac {(n+1)(n)(n-1)(n-2)}{4} + 3frac{n(n+1)(2n+1)}{6} - 2frac {n(n+1)}{2}$$
                                                                                                                                                Generating common denominators and with a bit of algebra we now have
                                                                                                                                                $$ frac {n^4-2n^3-n^2+2n+4n^3+6n^2+2n-4n^2-4n}{4}$$
                                                                                                                                                Combining like-terms we have reached our solution:
                                                                                                                                                $$ frac {n^4+2n^3+n^2}{4} = biggl(sum_{k=0}^n kbiggr)^2$$






                                                                                                                                                share|cite|improve this answer











                                                                                                                                                $endgroup$


















                                                                                                                                                  0












                                                                                                                                                  $begingroup$

                                                                                                                                                  We begin by writing $k^3$ in a more clever fashion: $k^3 = k(k-1)(k-2) + 3k^2 - 2k$ :



                                                                                                                                                  $$sum_{k=0}^n k^3 = sum_{k=0}^n k(k-1)(k-2) + 3k^2 -2k$$



                                                                                                                                                  Distributing the summation we obtain: $$ sum_{k=0}^n k(k-1)(k-2) + sum_{k=0}^n 3k^2 - sum_{k=0}^n 2k$$



                                                                                                                                                  Notice, $$k(k-1)(k-2) = frac {k!}{(k-3)!}$$
                                                                                                                                                  Now we have
                                                                                                                                                  $$sum_{k=0}^n frac {k!}{(k-3)!} + sum_{k=0}^n 3k^2 - sum_{k=0}^n 2k$$
                                                                                                                                                  Notice that we have nearly obtained the binomial expansion of K choose 3, all we need to do is divide by 3! So we offset this by also taking the product of 3!
                                                                                                                                                  $$sum_{k=0}^n frac {k!}{(k-3)!} = 3!sum_{k=0}^n frac {k!}{(k-3)!3!} = 3!sum_{k=0}^nbinom{k}{3} = 3!binom{n+1}{4}$$
                                                                                                                                                  We have now obtained
                                                                                                                                                  $$sum_{k=0}^n k^3 = 3!binom{n+1}{4} + 3sum_{k=0}^n k^2 - 2sum_{k=0}^n k$$
                                                                                                                                                  Focusing solely on the right-hand side we have
                                                                                                                                                  $$6biggl(frac {(n+1)!}{(n-3)!4!}biggr) + 3sum_{k=0}^n k^2 - 2sum_{k=0}^n k$$
                                                                                                                                                  Assuming we already know the sum of the sequence of integers and squared integers (the 2 sums we have left) we have
                                                                                                                                                  $$ frac {(n+1)(n)(n-1)(n-2)}{4} + 3frac{n(n+1)(2n+1)}{6} - 2frac {n(n+1)}{2}$$
                                                                                                                                                  Generating common denominators and with a bit of algebra we now have
                                                                                                                                                  $$ frac {n^4-2n^3-n^2+2n+4n^3+6n^2+2n-4n^2-4n}{4}$$
                                                                                                                                                  Combining like-terms we have reached our solution:
                                                                                                                                                  $$ frac {n^4+2n^3+n^2}{4} = biggl(sum_{k=0}^n kbiggr)^2$$






                                                                                                                                                  share|cite|improve this answer











                                                                                                                                                  $endgroup$
















                                                                                                                                                    0












                                                                                                                                                    0








                                                                                                                                                    0





                                                                                                                                                    $begingroup$

                                                                                                                                                    We begin by writing $k^3$ in a more clever fashion: $k^3 = k(k-1)(k-2) + 3k^2 - 2k$ :



                                                                                                                                                    $$sum_{k=0}^n k^3 = sum_{k=0}^n k(k-1)(k-2) + 3k^2 -2k$$



                                                                                                                                                    Distributing the summation we obtain: $$ sum_{k=0}^n k(k-1)(k-2) + sum_{k=0}^n 3k^2 - sum_{k=0}^n 2k$$



                                                                                                                                                    Notice, $$k(k-1)(k-2) = frac {k!}{(k-3)!}$$
                                                                                                                                                    Now we have
                                                                                                                                                    $$sum_{k=0}^n frac {k!}{(k-3)!} + sum_{k=0}^n 3k^2 - sum_{k=0}^n 2k$$
                                                                                                                                                    Notice that we have nearly obtained the binomial expansion of K choose 3, all we need to do is divide by 3! So we offset this by also taking the product of 3!
                                                                                                                                                    $$sum_{k=0}^n frac {k!}{(k-3)!} = 3!sum_{k=0}^n frac {k!}{(k-3)!3!} = 3!sum_{k=0}^nbinom{k}{3} = 3!binom{n+1}{4}$$
                                                                                                                                                    We have now obtained
                                                                                                                                                    $$sum_{k=0}^n k^3 = 3!binom{n+1}{4} + 3sum_{k=0}^n k^2 - 2sum_{k=0}^n k$$
                                                                                                                                                    Focusing solely on the right-hand side we have
                                                                                                                                                    $$6biggl(frac {(n+1)!}{(n-3)!4!}biggr) + 3sum_{k=0}^n k^2 - 2sum_{k=0}^n k$$
                                                                                                                                                    Assuming we already know the sum of the sequence of integers and squared integers (the 2 sums we have left) we have
                                                                                                                                                    $$ frac {(n+1)(n)(n-1)(n-2)}{4} + 3frac{n(n+1)(2n+1)}{6} - 2frac {n(n+1)}{2}$$
                                                                                                                                                    Generating common denominators and with a bit of algebra we now have
                                                                                                                                                    $$ frac {n^4-2n^3-n^2+2n+4n^3+6n^2+2n-4n^2-4n}{4}$$
                                                                                                                                                    Combining like-terms we have reached our solution:
                                                                                                                                                    $$ frac {n^4+2n^3+n^2}{4} = biggl(sum_{k=0}^n kbiggr)^2$$






                                                                                                                                                    share|cite|improve this answer











                                                                                                                                                    $endgroup$



                                                                                                                                                    We begin by writing $k^3$ in a more clever fashion: $k^3 = k(k-1)(k-2) + 3k^2 - 2k$ :



                                                                                                                                                    $$sum_{k=0}^n k^3 = sum_{k=0}^n k(k-1)(k-2) + 3k^2 -2k$$



                                                                                                                                                    Distributing the summation we obtain: $$ sum_{k=0}^n k(k-1)(k-2) + sum_{k=0}^n 3k^2 - sum_{k=0}^n 2k$$



                                                                                                                                                    Notice, $$k(k-1)(k-2) = frac {k!}{(k-3)!}$$
                                                                                                                                                    Now we have
                                                                                                                                                    $$sum_{k=0}^n frac {k!}{(k-3)!} + sum_{k=0}^n 3k^2 - sum_{k=0}^n 2k$$
                                                                                                                                                    Notice that we have nearly obtained the binomial expansion of K choose 3, all we need to do is divide by 3! So we offset this by also taking the product of 3!
                                                                                                                                                    $$sum_{k=0}^n frac {k!}{(k-3)!} = 3!sum_{k=0}^n frac {k!}{(k-3)!3!} = 3!sum_{k=0}^nbinom{k}{3} = 3!binom{n+1}{4}$$
                                                                                                                                                    We have now obtained
                                                                                                                                                    $$sum_{k=0}^n k^3 = 3!binom{n+1}{4} + 3sum_{k=0}^n k^2 - 2sum_{k=0}^n k$$
                                                                                                                                                    Focusing solely on the right-hand side we have
                                                                                                                                                    $$6biggl(frac {(n+1)!}{(n-3)!4!}biggr) + 3sum_{k=0}^n k^2 - 2sum_{k=0}^n k$$
                                                                                                                                                    Assuming we already know the sum of the sequence of integers and squared integers (the 2 sums we have left) we have
                                                                                                                                                    $$ frac {(n+1)(n)(n-1)(n-2)}{4} + 3frac{n(n+1)(2n+1)}{6} - 2frac {n(n+1)}{2}$$
                                                                                                                                                    Generating common denominators and with a bit of algebra we now have
                                                                                                                                                    $$ frac {n^4-2n^3-n^2+2n+4n^3+6n^2+2n-4n^2-4n}{4}$$
                                                                                                                                                    Combining like-terms we have reached our solution:
                                                                                                                                                    $$ frac {n^4+2n^3+n^2}{4} = biggl(sum_{k=0}^n kbiggr)^2$$







                                                                                                                                                    share|cite|improve this answer














                                                                                                                                                    share|cite|improve this answer



                                                                                                                                                    share|cite|improve this answer








                                                                                                                                                    edited Aug 18 '18 at 18:18









                                                                                                                                                    tinlyx

                                                                                                                                                    95421118




                                                                                                                                                    95421118










                                                                                                                                                    answered Aug 18 '18 at 17:16









                                                                                                                                                    Alexander B.Alexander B.

                                                                                                                                                    11




                                                                                                                                                    11






























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