Question

Let $f\left(x\right)$ be a function such that; $f(x-1)+f(x+1)=\sqrt{3}f\left(x\right),\forall x\in R$. If $f\left(5\right)=100$, then find $\sum _{r=0}^{99}f(5+12r)$

Answer 1

Assume $f\left(4\right)=a$, $f\left(5\right)=100$.
$f\left(6\right)=100\sqrt{3}-a$
$f\left(7\right)=f\left(6\right)-f\left(5\right)=200-\sqrt{3}a$
$f\left(8\right)=f\left(7\right)-f\left(6\right)=100\sqrt{3}-2a$
$f\left(9\right)=100-\sqrt{3}a$
$f\left(10\right)=-a$
$f\left(x\right)=-f(x+6)$
$f(x+12)=f\left(x\right)$
Period is $12$.
Sum of series$=100\times 100=10000$