Question

The coefficient of $x^9$ in $(x+2)(x+4)(x+8).......(x+1024)$ is

• A. $2046$
• B. $1023$
• C. $55$
• D. $0$

Answer 1

The correct option is A $2046$

The give expression can be written as
$(x+2)(x+2^2)(x+2^3)..(x+2^{10})$
$=x^{10}+x^9(2+2^2+2^3+..{.2}^{10})....$
Hence the coefficient of $x^9$ is
$=2+2^2+2^3+..{.2}^{10}$.
This is a G.P
The sum of the G.P is
$=\frac{2(2^{10}-1)}{2-1}$
$=2(2^{10}-1)$
$=2^{11}-2$
$=2048-2$
$=2046$