Question

The coefficient of $x^{2^{n+1}}$ in $\frac{1}{(1+x)(1+x^2)(1+x^4)....(1+x^{2^n})}$ (when expanded in ascending power of x), where $|x|<1$, is

• A. $1$
• B. $-1$
• C. $2$
• D. $n$
• E. None of these.

Answer 1

Answer: (A)

$1$

$\frac{1}{(1+x)(1+x^2)(1+x^4)....(1+x^{2^n})}=\frac{1-x}{(1-x)(1+x)(1+x^2)...(1+x^{2^n})}=\frac{1-x}{1-x^{2^{n+1}}}$
$=(1-x){(1-x^{2^{n+1}})}^{-1}$
$=(1-x)\{1+x^{2^{n+1}}+{\left(x^{2^{n+1}}\right)}^2+....\}$
Hence the coefficient of $x^{2^{n+1}}=1$