Question

The period of $f\left(x\right)=\sin \frac{\pi x}{n!}-\cos \frac{\pi x}{(n+1)!}$

• A. $(n+1)!$
• B. $2(n+1)!$
• C. $2n!$
• D. $\frac{(n+1)}{2!}$

Answer 1

Answer: (B)

$2(n+1)!$

Let $f\left(x\right)=\sin \frac{\pi x}{n!}-\cos \frac{\pi x}{(n+1)!}$
Period of $\sin \frac{\pi x}{n!}=\frac{2\pi }{\pi }\times n!=2n!$
Period of $\cos \frac{\pi x}{(n+1)!}=\frac{2\pi }{\pi }(n+1)!=2(n+1)!$
Period of $f\left(x\right)=LCMof(2n!,2(n+1)!)$
$=2(n+1)!$