AoPS - Problem

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Problem

Source: Kazakhstan NMO 2016, P6

Tags: number theory, algebra

rightways

17.03.2016 11:31

Given a strictly increasing infinite sequence $\{a_n\}$ of positive real numbers such that for any $n\in N$ : $a_{n+2}=(a_{n+1}-a_{n})^{\sqrt{n}}+n^{-\sqrt{n}}$ Prove that for any $C>0$ there exist a positive integer $m(C)$ (depended on $C$ ) such that $a_{m(C)}>C$ .

dgrozev

20.03.2016 19:20

In other words, one have to prove $a_n$ is not bounded. Suppose on the contrary, the sequence is bounded. Then, since it's increasing, it has a limit, say $\ell>0$ . Take a limit of both side of the equality, and since $\lim_{n\to\infty}(a_{n+1}-a_n)=0$ , we get: $\ell=0$ a contradiction.