Problem

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Tags: inequalities, induction, algebra proposed, algebra



Let $n \ge 2$ be integer. Let $a_0$, $a_1$, ... $a_n$ be sequence of positive reals such that: $(a_{k-1}+a_k)(a_k+a_{k+1})=a_{k-1}-a_{k+1}$, for $k=1, 2, ..., n-1$. Prove $a_n< \frac{1}{n-1}$.