Problem

Source: Indonesia IMO 2010 TST, Stage 1, Test 3, Problem 3

Tags: inequalities, induction, algebra proposed, algebra



Let $ a_1,a_2,\dots$ be sequence of real numbers such that $ a_1=1$, $ a_2=\dfrac{4}{3}$, and \[ a_{n+1}=\sqrt{1+a_na_{n-1}}, \quad \forall n \ge 2.\] Prove that for all $ n \ge 2$, \[ a_n^2>a_{n-1}^2+\dfrac{1}{2}\] and \[ 1+\dfrac{1}{a_1}+\dfrac{1}{a_2}+\dots+\dfrac{1}{a_n}>2a_n.\] Fajar Yuliawan, Bandung