An Integer Hidden in a Recursion

★ 8/10number theory Recursion sequences Induction

Problem

A sequence $\{a_n\}$ satisfies $a_1 = 1$ and

a_{n+1} = \frac{1}{2}a_n + \frac{1}{a_n}.

Prove that for every $n \geqslant 2$ ,

\frac{2}{\sqrt{a_n^2 - 2}}

is a positive integer.

Answer

Solution

Difficulty	8/10
Topics	number theory, Recursion, sequences, Induction

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