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Show that 0 ≤ f(n+1) - f(n) ≤ 1 and find n s.t. f(n) = 1025

Source: Canada National Mathematical Olympiad 1990 - Problem 5

October 4, 2011
functionalgebrastrong inductionnumber theoryfunctional equation

Problem Statement

The function f:N→Rf : \mathbb N \to \mathbb R satisfies f(1)=1,f(2)=2f(1) = 1, f(2) = 2 and f(n+2)=f(n+2−f(n+1))+f(n+1−f(n)).f (n+2) = f(n+2 - f(n+1) ) + f(n+1 - f(n) ). Show that 0≤f(n+1)−f(n)≤10 \leq f(n+1) - f(n) \leq 1. Find all nn for which f(n)=1025f(n) = 1025.
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