Mathematical Induction Prove the Formula
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Given a positive number $a$ and a positive integer $k$. Let the sequence $x_n$ be given recurrently: $x_1 = a^{frac{1}{k}}$, $x_{n + 1} = frac{a}{x_n} ^ {frac{1}{k}}$. Prove that the general term formula has the form $x_n = a^{frac{1 - (- k) ^{- n}}{k + 1}}$
induction
asked Nov 23 at 12:57
sclerd
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Given a positive number $a$ and a positive integer $k$. Let the sequence $x_n$ be given recurrently: $x_1 = a^{frac{1}{k}}$, $x_{n + 1} = frac{a}{x_n} ^ {frac{1}{k}}$. Prove that the general term formula has the form $x_n = a^{frac{1 - (- k) ^{- n}}{k + 1}}$
induction
asked Nov 23 at 12:57
sclerd
11
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Given a positive number $a$ and a positive integer $k$. Let the sequence $x_n$ be given recurrently: $x_1 = a^{frac{1}{k}}$, $x_{n + 1} = frac{a}{x_n} ^ {frac{1}{k}}$. Prove that the general term formula has the form $x_n = a^{frac{1 - (- k) ^{- n}}{k + 1}}$
induction
asked Nov 23 at 12:57
sclerd
11
Given a positive number $a$ and a positive integer $k$. Let the sequence $x_n$ be given recurrently: $x_1 = a^{frac{1}{k}}$, $x_{n + 1} = frac{a}{x_n} ^ {frac{1}{k}}$. Prove that the general term formula has the form $x_n = a^{frac{1 - (- k) ^{- n}}{k + 1}}$
induction
induction
asked Nov 23 at 12:57
sclerd
11
asked Nov 23 at 12:57
sclerd
11
asked Nov 23 at 12:57
sclerd
11
asked Nov 23 at 12:57
sclerd
11
asked Nov 23 at 12:57
sclerd
11
11
add a comment |
add a comment |
1 Answer
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- The formula is valid for $n=1$.
- Assuming it is valid for $n$, calculate
$$x_{n+1}=left(frac a{x_n}right)^{1/k}=left(a^{1-frac{1-(-k)^n}{k+1}}right)^{1/k}=dots$$
answered Nov 23 at 15:41
Berci
59.4k23672
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1 Answer
1
active
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
- The formula is valid for $n=1$.
- Assuming it is valid for $n$, calculate
$$x_{n+1}=left(frac a{x_n}right)^{1/k}=left(a^{1-frac{1-(-k)^n}{k+1}}right)^{1/k}=dots$$
answered Nov 23 at 15:41
Berci
59.4k23672
add a comment |
up vote
0
down vote
- The formula is valid for $n=1$.
- Assuming it is valid for $n$, calculate
$$x_{n+1}=left(frac a{x_n}right)^{1/k}=left(a^{1-frac{1-(-k)^n}{k+1}}right)^{1/k}=dots$$
answered Nov 23 at 15:41
Berci
59.4k23672
add a comment |
up vote
0
down vote
up vote
0
down vote
- The formula is valid for $n=1$.
- Assuming it is valid for $n$, calculate
$$x_{n+1}=left(frac a{x_n}right)^{1/k}=left(a^{1-frac{1-(-k)^n}{k+1}}right)^{1/k}=dots$$
answered Nov 23 at 15:41
Berci
59.4k23672
- The formula is valid for $n=1$.
- Assuming it is valid for $n$, calculate
$$x_{n+1}=left(frac a{x_n}right)^{1/k}=left(a^{1-frac{1-(-k)^n}{k+1}}right)^{1/k}=dots$$
answered Nov 23 at 15:41
Berci
59.4k23672
answered Nov 23 at 15:41
Berci
59.4k23672
answered Nov 23 at 15:41
Berci
59.4k23672
answered Nov 23 at 15:41
Berci
59.4k23672
59.4k23672
add a comment |
add a comment |
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