Question

The function $f(x)=\sin \frac{\pi x}{n!}-\cos \frac{\pi x}{(n+1)!}$ is

• A. non periodic
• B. periodic with period 2(n!)
• C. periodic with period $2(n+1)$ !
• D. periodic with period $n$ !

Answer 1

Answer: (C)

Since the period of $\sin x$ is $2\pi$ so the period of $\sin \frac{\pi x}{n!}$ is $\frac{2\pi n!}{\pi }=2(n!)$.
The period of $\cos \frac{\pi x}{(n+1)!}$ is $2(n+1)$ ! The period of $f(x)=1.c.m.(2(n!),2(n+1)!)=$ $2(n+1)$ !