Question

The coefficient of ${(p+2)}^{th}$ term from the end in the expansion of ${(x-1/x)}^{2n+1}$ is

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

Answer 1

The correct option is B $\frac{{(-1)}^p(2n+1)!}{(2n-p)!(1+p)!}$

For term from the end, we write the expression as
$-(\frac{1}{x}-x{)}^{2n+1}$
Hence the coefficient of
$T_{p+2}$
$=-\left[\right(-1{)}^{p+1}{}^{2n+1}{C}_{p+1}$
$=-\left[\right(-1{)}^{p+1}\frac{(2n+1)!}{(2n-p)!(p+1)!}]$
$=(-1{)}^p\frac{(2n+1)!}{(2n-p)!(p+1)!}$