Question

The coefficient of $x^n$ in the product $(1-x)(1-2x)(1-2^2.x)(1-2^3.x)...(1-2^n.x)$ is equal

• A. $(1-2^{n+1}){.2}^{\frac{n(n-1)}{2}}$
• B. $(2^{n+1}-1){.2}^{\frac{n(n-1)}{2}}$
• C. $(1-2n){.2}^{\frac{n(n-1)}{2}}$
• D. $noneofthese$

Answer 1

The correct option is C $(1-2n){.2}^{\frac{n(n-1)}{2}}$

$(1-x)(1-2x)(1-2^2.x)(1-2^3.x)...(1-2^n.x)$
Take n = 1
$\therefore (1-x)$
Coefficient of $x^n=x^1$ is -1.
Substitute n=1, in the options
a. $(1-2^{n+1}){.2}^{\frac{n(n-1)}{2}}=(1-2^{n+1})2^{\frac{1(1-1)}{2}}=(1-4)2^{\circ }=3$
b. $(2^{n+1}-1){.2}^{\frac{n(n-1)}{2}}=(2^2-1)2^{\frac{1(1-1)}{2}}=3\times 2^{\circ }=3$
c. $(1-2n){.2}^{\frac{n(n-1)}{2}}=(1-2(1\left)\right)2^{\frac{1(1-1)}{2}}=-1\times 2^{\circ }=-1$