Question

Find the period of the real-valued function satisfying, $f\left(x\right)+f(x+4)=f(x+2)+f(x+6)$

Answer 1

Given $f\left(x\right)+f(x+4)=f(x+2)+f(x+6)$ ....(1)
Replacing $x$ by $x-2$
$f(x-2)+f(x+2)=f\left(x\right)+f(x+4)$ ....(2)
Adding (1) and (2), we get
$f(x-2)=f(x+6)$
Replacing $x$ by $x+2$, we get
$f\left(x\right)=f(x+8)$
Hence, period is $8$