Question

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

Accepted Answer

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

Periodicity of a functional equation

Explanation

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