2020 APMO 第 2 题

Show that $r=2$ is the largest real number $r$ which satisfies the following condition:

If a sequence $a_{1}, a_{2}, \ldots$ of positive integers fulfills the inequalities

$$a_{n} \leq a_{n+2} \leq \sqrt{a_{n}^{2}+r a_{n+1}}$$

for every positive integer $n$, then there exists a positive integer $M$ such that $a_{n+2}=a_{n}$ for every $n \geq M$.

证明$r=2$是满足以下条件的最大实数$r$：

如果正整数序列 $a_{1}, a_{2}, \ldots$ 满足不等式

$$a_{n} \leq a_{n+2} \leq \sqrt{a_{n}^{2}+r a_{n+1}}$$

对于每个正整数$n$，则存在一个正整数$M$，使得对于每个$n \geq M$，$a_{n+2}=a_{n}$。