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Easy sequence- Serbia Mathematical Olympiad 2011- Problem 1

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April 8, 2011
inequalitiesinductionalgebra proposedalgebra

Problem Statement

Let n≥2n \ge 2 be integer. Let a0a_0, a1a_1, ... ana_n be sequence of positive reals such that: (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 k=1,2,...,n−1k=1, 2, ..., n-1. Prove an<1n−1a_n< \frac{1}{n-1}.
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