Mind Warmups-11

Calculus Level 4

Let g ( x ) g(x) be a function satisfying g ( x + 1 ) + g ( x − 1 ) = g ( x ) g(x+1)+g(x-1)=g(x) . What is the period of g g ?

g g is not periodic. 3 6 2

Sanjeet Raria by Sanjeet Raria , 27, India

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1 solution

Chew-Seong Cheong
Sep 21, 2018

Given that

g ( x + 1 ) + g ( x − 1 ) = g ( x ) ⟹ g ( x + 1 ) = g ( x ) − g ( x − 1 ) g ( x + 2 ) = g ( x + 1 ) − g ( x ) = g ( x ) − g ( x − 1 ) − g ( x ) ⟹ g ( x + 2 ) = − g ( x − 1 ) g ( x + 2 + 3 ) = − g ( x − 1 + 3 ) g ( x + 5 ) = − g ( x + 2 ) = g ( x − 1 ) ⟹ g ( x + 6 ) = g ( x ) \begin{aligned} g(x+1) + g(x-1) & = g(x) \\ \implies g(x+1) & = g(x) - g(x-1) \\ g(x+2) & = g(x+1) - g(x) \\ & = g(x) - g(x-1) - g(x) \\ \implies g(x+2) & = - g(x-1) \\ g(x+2+{\color{#D61F06}3}) & = - g(x-1+{\color{#D61F06}3}) \\ g(x+5) & = - g(x+2) = g(x-1) \\ \implies g(x + {\color{#3D99F6}6}) & = g(x) \end{aligned}

Therefore, the period of g g is 6 \boxed 6 .

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