Question

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

Answer 1

Find the value of $n$:

$(1+x+x^2+x^3\dots .+x^{2n})(1-x+x^2-x^3\dots .+x^{2n})=a_0+a_{1}x+a_2{x}^2+.....$

The sum of coefficient of all even powers $\begin{array}{rcl}& =& \left(a_0+a_2+a_4+.....\right)\end{array}$

By putting $x=1$, we get

$2n+1=a_0+a_1+a_2+........\left(1\right)$

By putting $x=-1$, we get

$2n+1=a_0-a_1+a_2+........\left(2\right)$

By adding $\left(1\right)\mathrm{and}\left(2\right)$, we get

$\begin{array}{rcl}2\left(2n+1\right)& =& 2\left(a_0+a_2+a_4+.....\right)\\ & \Rightarrow & 2n+1=61\\ \Rightarrow n& =& 30\end{array}$

Hence, the value of $n$ is $30$.