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Rather known sequence

Source: Romanian IMO TST 2006, day 2, problem 1

April 22, 2006
inductionquadraticsalgebraalgebra proposed

Problem Statement

Let {an}n≥1\{a_n\}_{n\geq 1} be a sequence with a1=1a_1=1, a2=4a_2=4 and for all n>1n>1, an=an−1an+1+1. a_{n} = \sqrt{ a_{n-1}a_{n+1} + 1 } . a) Prove that all the terms of the sequence are positive integers. b) Prove that 2anan+1+12a_na_{n+1}+1 is a perfect square for all positive integers nn. Valentin Vornicu
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