Question

The coefficient of $x^{2}in(1+x)(1+2x)(1+4x)....(1+2^{n}x)$ is

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

Coefficient of $x^2=\frac{1}{2}\left[\right(1+2+2^2....+2^n{)}^2]-[1^2+2^2+...(2^n{)}^2]=\frac{1}{2}[{\left(\frac{2^{n+1}-1}{1}\right)}^2-\left(\frac{(2^2{)}^{n+1}-1}{3}\right)]=\frac{1}{2}\left[\right(2^{n+1}\left)\left[\right(2^{n+1}-1)-\frac{2^{n+1}+1}{3}]\right]=\frac{(2^{n+1}-1)(2^{n+1}-2)}{3}$