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

Let,$\left(1+x+x^2+\mathrm{.....}+x^{2n}\right)\left(1-x+x^2-x^3+\mathrm{.....}+x^{2n}\right)=a_0+a_{1}x+a_2{x}^2+\mathrm{....}$

Put x = 1

$1\left(2n+1\right)=a_0+a_1+a_2+\mathrm{....}+a_{4n}$….(i)

Put x = -1

$\left(2n+1\right)\times 1=a_0-a_1+a_2-\mathrm{...}+a_{4n}$..(ii)

From (i) + (ii)

$4n+2=2\left(a_0+a_2+\mathrm{...}\right)$

           $=2\times 61$

$\Rightarrow 2n+1=61\Rightarrow n=30$