RS-SMO 2011 Problem 1

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 $a_n< \frac{1}{n-1}$ .