Question

The period of the function
$f\left(x\right)=cos\left(\frac{\Pi x}{n!}\right)-sin\left(\frac{\Pi x}{(n+1)!}\right)$ is.

• A. $2(n+1)!$
• B. $2(n!)$
• C. $(n+1)$
• D. not periodic

Answer 1

The correct option is A $2(n+1)!$

$f\left(x\right)=\cos \left(\frac{\Pi x}{n!}\right)-\sin \left(\frac{\Pi x}{(n+1)!}\right)$
Period of $\cos \left(\frac{\Pi x}{n!}\right)$ is $2(n!)$
Period of $\sin \left(\frac{\Pi x}{(n+1)!}\right)$ is $2(n+1)!$
$\therefore$ Period of $f\left(x\right)$ is L.C.M of $2(n!)$ and $2(n+1)!$ $=2(n+1)!$
Hence, option A.