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.

$$f(x+1) +f(x-1)=\sqrt3 f(x)$$
Let $x=x+1$ and $x=x-1$
$$f(x+2) + f(x)=\sqrt3 f(x+1)$$
$$f(x) + f(x - 2)=\sqrt3 f(x-1)$$
Adding the above we get 
$$f(x+2) + f(x - 2)+ 2f(x)=\sqrt3(f(x+1) +f(x-1))$$
$$=3f(x)$$
$$f(x+2) + f(x - 2)=f(x)$$
Letting $x=x+2$ in above 
$$f(x+4) + f(x)=f(x+2)=f(x)-f(x-2)$$
