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

Let $\left(1 +x + x^{2}+ ... +x^{2n}\right) \left( 1-x+x^{2}-x^{3}+ ... + x^{2n}\right)$
$= a_{0} + a_{1}x + a_{2}x^{2} + a_{3}x^{3} + a_{4}x^{4} +\ldots+a_{4n}x^{4n}$
So,
$a_{0} + a_{1} + a_{2} +\ldots+ a_{4n} = 2n + 1\quad\ldots\left(1\right)$
$a_{0} - a_{1} + a_{2} - a_{3}\ldots+a_{4n} = 2n + 1\quad \ldots\left(2\right)$
$\Rightarrow a_{0} + a_{2} + a_{4} +\ldots+a_{4n} = 2n + 1$
$\Rightarrow 2n + 1 = 61\quad\quad\Rightarrow n = 30$