If x r occurs in the expansion of ( x + 1 x ) n , then its coefficient is

Question

If $x^{r}$ occurs in the expansion of ${(x + \frac{1}{x})}^{n},$ then its coefficient is

Smart Achievers · Accepted Answer

(3)

$T_{m + 1} = {}^{n}C_{m} {(x)}^{n - m} {(\frac{1}{x})}^{m} = {}^{n}C_{m} x^{n - 2 m}$

$If x^{r} occur in expansion, then$

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

$Coefficient of x^{r} = {}^{n}C_{(\frac{n - r}{2})} = \frac{n!}{(\frac{n - r}{2})! (\frac{n + r}{2})!}$

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