sequence and inequality
Source: Indonesia IMO 2010 TST, Stage 1, Test 3, Problem 3
November 12, 2009
inequalitiesinductionalgebra proposedalgebra
Problem Statement
Let be sequence of real numbers such that a_1\equal{}1, a_2\equal{}\dfrac{4}{3}, and a_{n\plus{}1}\equal{}\sqrt{1\plus{}a_na_{n\minus{}1}}, \forall n \ge 2. Prove that for all , a_n^2>a_{n\minus{}1}^2\plus{}\dfrac{1}{2} and 1\plus{}\dfrac{1}{a_1}\plus{}\dfrac{1}{a_2}\plus{}\dots\plus{}\dfrac{1}{a_n}>2a_n.
Fajar Yuliawan, Bandung