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 $f(x)=1 . c . m .(2(n !), 2(n+1) !)=$ $2(n+1)$ !