Question

If the sum of the coefficients of all even powers of x in the product
$\left(1+x+x^2+...+x^{2n}\right)\left(1-x+x^2-x^3+...+x^{2n}\right)$
is 61, then n is equal to __________.
Given 15

Answer 1

Answer: 30.00

Let $\left(1+x+x^2+...+x^{2n}\right)\left(1-x+x^2-x^3+...+x^{2n}\right)$
$=a_0+a_{1}x+a_2{x}^2+a_3{x}^3+a_4{x}^4+\dots +a_{4n}{x}^{4n}$
So,
$a_0+a_1+a_2+\dots +a_{4n}=2n+1\dots \left(1\right)$
$a_0-a_1+a_2-a_3\dots +a_{4n}=2n+1\dots \left(2\right)$
$\Rightarrow a_0+a_2+a_4+\dots +a_{4n}=2n+1$
$\Rightarrow 2n+1=61\Rightarrow n=30$