<s>[INST] If $f(x+1)+f(x-1)=\sqrt 3 f(x), \forall x$ then $f$ is periodic. 
If $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
  $f(x+1)+f(x-1)=\sqrt 3 f(x), \forall x$ then $f$ is periodic.


I tried to replace $x$ by $x+1, x-1$ in the equality,to get something like $f(x + k)=f(x)$ but without success.
Any help is appreciated.
 [/INST] From the given equation, $$f(x+4)=\sqrt{3}f(x+3)-f(x+2),\\\sqrt{3}f(x+3)=3f(x+2)-\sqrt{3}f(x+1),\\f(x+2)=\sqrt{3}f(x+1)-f(x).$$ Adding those, we obtain that $$f(x+4)=f(x+2)-f(x),$$ and also $$f(x+6)=f(x+4)-f(x+2),$$ therefore $f(x+6)=-f(x)$. Hence, $$f(x+12)=-f(x+6)=f(x),$$ therefore $f$ is periodic with period $12$.
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