Question

If the sum of the coefficients of all even powers of $x$ in the product $(1+x+x^2+x^3+....+x^{2n})(1-x+x^2-x^3....+x^{2n})$ is $61$, then $n$ is equal to

Answer 1

Let $(1+x+x^2+...+x^{2n})(1-x+x^2-...+x^{2n})=a_0+a_{1}x+a_2{x}^2+...$
Put $x=1$
$2n+1=a_0+a_1+a_2+a_3+...\left(1\right)$

Put $x=-1$
$2n+1=a_0-a_1+a_2-a_3+...\left(2\right)$
Adding (1) and (2)
$2(2n+1)=2(a_0+a_2+a_4+...$
$\Rightarrow 2n+1=61$
$\Rightarrow n=30$