Question

Find the coefficient of $x^r$ in the expansion of ${(x^2+\frac{1}{x^2})}^n$.

Answer 1

General term in expansion of ${(x^2+\frac{1}{x^2})}^n$ is given by ${}^n{C}_p{x}^{2p}\frac{1}{x^{2n-2p}}{\Rightarrow }^n{C}_p{x}^{4p-2n}$
To find coefficient of $x^r$, put $r=4p-2n\Rightarrow p=\frac{r+2n}{4}$
We get the answer ${}^n{C}_{\frac{r+2n}{4}}$