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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 $x_n = a^{\frac{1 - (- k) ^{- n}}{k + 1}}$

sclerd
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1 Answers1

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  1. The formula is valid for $n=1$.
  2. Assuming it is valid for $n$, calculate $$x_{n+1}=\left(\frac a{x_n}\right)^{1/k}=\left(a^{1-\frac{1-(-k)^n}{k+1}}\right)^{1/k}=\dots$$
Berci
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