Question

Find the term of expansion of ${(x+\frac{1}{x})}^n$ which does not contain $x$.

Answer 1

Note:- The general term or $(r+1{)}^{th}$ term of $(a+b{)}^n$ is $T_{r+1}{=}^n{C}_r{a}^{n-r}{b}^r$

$T_{r+1}{=}^r{C}_r{\left(x\right)}^{n-r}{\left(1/x\right)}^r$

$\Rightarrow T_{r+1}{=}^n{C}_r\frac{x^{n-r}}{n^r}$

$\Rightarrow T_{r+1}{=}^n{C}_r{x}^{n-r-r}{=}^n{C}_r{x}^{n-2r}$

$\therefore x^{n-2r}=x^0$

$\Rightarrow n-2r=0$

$\Rightarrow n=2r$

$\Rightarrow \frac{n}{2}=r$

Therefore $T_{r+1}=T_{\frac{n+2}{2}}$ is the term which does not contains $x$