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a_n < 1/(n-1) if (a_{k-1}+a_{k})(a_{k}+a_{k+1})=a_{k-1}-a_{k+1}

Source: 2011 Grand Duchy of Lithuania, Mathematical Contest p2 (Baltic Way TST)

October 3, 2020
algebrarecurrence relationinequalities

Problem Statement

Let n≥2n \ge 2 be a natural number and suppose that positive numbers a0,a1,...,ana_0,a_1,...,a_n satisfy the equality (ak−1+ak)(ak+ak+1)=ak−1−ak+1(a_{k-1}+a_{k})(a_{k}+a_{k+1})=a_{k-1}-a_{k+1} for each k=1,2,...,n−1k =1,2,...,n -1. Prove that an<1n−1a_n< \frac{1}{n-1}
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