Question

The coefficient of $x^n$ in $\frac{(1+x{)}^2}{(1-x{)}^3}$ is :

• A. $3n^2$+2n+1
• B. $2n^2$+2n+1
• C. $n^2$+n+1
• D. $2n^2$+2n+1

Answer 1

The correct option is B

$2n^2$+2n+1

$(1+x{)}^2(1-x{)}^{-3}=(1+2x+x^2)(1-x{)}^{-3}$

$Coeff.of{x}^n=(Coeff.of{x}^n+2.coeff.x^{n-1}+$

$coeff.of{x}^{n-2}\left)in\right(1-x{)}^{-3}$ .

$={}^{n+2}{C}_3+2.{}^{n+1}{C}_2+{}^n{C}_2$

=[(n+2) (n+1)+2(n+1)n+n(n-1)]/2

$=2n^2+2n+1$