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Periodicity of a functional equation

CAT 2005 Slot 1 · Quantitative Ability · Hard · Algebra

This is a hard Quantitative Ability question from the CAT 2005 Slot 1 paper. It tests Algebra. The full answer key and a step-by-step explanation are below — try it yourself first, then reveal the solution.

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Let g(x)g(x) be a function such that g(x+1)+g(x−1)=g(x)g(x + 1) + g(x - 1) = g(x) for every real xx. Then for what value of pp is the relation g(x+p)=g(x)g(x + p) = g(x) necessarily true for every real xx?

  1. A.

    5

  2. B.

    3

  3. C.

    2

  4. D.

    6

Answer

D

Explanation

Given: g(x+1)+g(x−1)=g(x)g(x+1) + g(x-1) = g(x). Replace xx with x+1x+1: g(x+2)+g(x)=g(x+1)g(x+2) + g(x) = g(x+1). Adding both equations: g(x+2)+g(x−1)=0  ⟹  g(x+3)=−g(x)g(x+2) + g(x-1) = 0 \implies g(x+3) = -g(x). Replacing xx with x+3x+3: g(x+6)=−g(x+3)=−(−g(x))=g(x)g(x+6) = -g(x+3) = -(-g(x)) = g(x). Therefore, the period p=6p = 6.

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