Mathematics - Binomial Theorem Question with Solution | TestHub

Coefficient of $x^2=\frac{1}{2}\left[{\left(1+2+2^2\dots \dots +2^n\right)}^2-\left[1^2+2^2+\dots .{\left(2^n\right)}^2\right]\right]$ $\begin{array}{rlrl}& =\frac{1}{2}\left[{\left(\frac{2^{n+1}-1}{1}\right)}^2-\left(\frac{{\left(2^2\right)}^{n+1}-1}{3}\right)\right]& =\frac{1}{2}\left[\left(2^{n+1}-1\right)\left[\left(2^{n+1}-1\right)-\frac{2^{n+1}+1}{3}\right]\right]& =\frac{\left(2^{n+1}-1\right)\left(2^{n+1}-2\right)}{3}\end{array}$