Question

Find the coefficient of $x^r$ in the expansion of $\frac{(1-x{)}^2}{(1+x{)}^3}$

Answer 1

The expression $=(1-4x+4x^2)(1+p_{1}x+p_2{x}^2+...+p_r{x}^r+...)$ suppose.
The coefficient of $x^r$ will be obtained by multiplying $p_r,pr-1,pr-2$ by $1,-4,4$ respectively, and adding the results ; hence
the required coefficient $=p_r-4p_{r-1}+4p_{r-2}$
But $p_r=(-1{)}^r\frac{(r+1)(r+2)}{2}$
Hence the required coefficient
$=(-1{)}^r\frac{(r+1)(r+2)}{2}-4(-1{)}^{r-1}\frac{r(r+1)}{2}+4(-1{)}^{r-2}\frac{(r-1)r}{2}$
$=\frac{(-1{)}^r}{2}\left[\right(r+1\left)\right(r+2)+4r(r+1)+4r(r-1\left)\right]$
$\frac{(-1{)}^r}{2}(9r^2+3r+2)$