Question

If f (x) +f (x+a) + f(x+2a) +....+ f (x+na) = constant; $\forall x$ $ϵR$ and $a>0$ and f(x) is periodic,then period of f(x), is

• A. (n+1) a
• B. $e^{n+1}a$
• C. na
• D. $e^{na}$

Answer 1

The correct option is A (n+1) a

$f\left(x\right)+f(x+a)+f(x+2a)+....+f(x+na)=k$
replacing $x=x+a$
$f(x+a)+f(x+2a)+....+f(x+(n+1\left)a\right)=k$
On subtracting second equation from first one -
$f\left(x\right)-f(x+(n+1\left)a\right)=0f\left(x\right)=f(x+(n+1)a\therefore T=(n+1)a$