Question

If $f\left(x\right)+f(x+2)=0,$ for all $x\in R$ then find the period of $f\left(x\right)$

Answer 1

As $f\left(x\right)+f(x+2)=0$ ......$\left(1\right)$

Replacing $x$ by then $f(x+2)+f(x+4)=0$ ...$\left(2\right)$

Subtracting $\left(2\right)$ from $\left(1\right)$, we get

$f(x+2)+f(x+4)-f\left(x\right)-f(x+2)=0$

$\Rightarrow f(x+4)-f\left(x\right)=0$

$\Rightarrow f(x+4)=f\left(x\right)$

Hence $f\left(x\right)$ has period $T=4$.