Question

Let $f\left(x\right)$ be periodic and $k$ be a positive real number such that $f(x+k)+f\left(x\right)=0$ for all $x\in$ $R$. If the period of $f\left(x\right)$ is $ak$. Find $a$

Answer 1

We have,$f(x+k)+f\left(x\right)=0,\forall x\in R$
$\Rightarrow f(x+k)=-f\left(x\right),\forall x\in R$, put $x=x+k$
$\Rightarrow f(x+2k)=-f(x+k),\forall x\in R$ [as $f(x+k)=-f\left(x\right)]$
$\Rightarrow f(x+2k)=f\left(x\right),\forall x\in R$ which clearly shows that $f\left(x\right)$ is periodic with period $2k$.
$a=2$