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Periodicity of function h satisfying (h(x-1)+h(x+1))/h(x)=k according to k

Source: Romanian District Olympiad 2016, Grade X, Problem 4

October 5, 2018

functionalgebrafunctional equation

Problem Statement

a) Prove that not all functions

f:\mathbb{R}\longrightarrow\mathbb{R}

that satisfy the equality f(x-1)+f(x+1) =\sqrt 5f(x) , \forall x\in\mathbb{R} , are periodic.
b) Prove that that all functions

g:\mathbb{R}\longrightarrow\mathbb{R}

that satisfy the equality g(x-1)+g(x+1)=\sqrt 3g(x) , \forall x\in\mathbb{R} , are periodic.