Question

If $|x|<1$, then the coefficient of $x^n$ in expansion of ${\left(1+x+x^2+x^3+\dots \right)}^2$ is:

• A. $n$
• B. $n-1$
• C. $n+2$
• D. $n+1$

Answer 1

Answer: (D)

$\because {\left(1+x+x^2+x^3+\dots \right)}^2={\left[(1-x{)}^{-1}\right]}^2$
$=(1-x{)}^{-2}$
Coefficient of $x^n$ in ${\left(1+x+x^2+\dots \right)}^2$
= coefficient of $x^n\text{ in }(1-x{)}^{-2}$
${=}^{n+2-1}{C}_{2-1}={}^{n+1}{C}_1$
$=n+1$