MathDB

inequality with functions

Source: Ireland 1994

June 29, 2009

inequalitiesfunctioninequalities proposed

Problem Statement

Let

f(n)

be defined for

n \in \mathbb{N}

by f(1)\equal{}2 and f(n\plus{}1)\equal{}f(n)^2\minus{}f(n)\plus{}1 for

n \ge 1

. Prove that for all

n >1:

1\minus{}\frac{1}{2^{2^{n\minus{}1}}}<\frac{1}{f(1)}\plus{}\frac{1}{f(2)}\plus{}...\plus{}\frac{1}{f(n)}<1\minus{}\frac{1}{2^{2^n}}