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

Question

If xr occurs in the expansion of ( x + (1/x) )n , then its coefficient is (A)  (n!/(r!)2) (B)  (n!/(r +1)! (r-1)!) (C) (n!/((n+r)2)!((n-r)2)! (D) (n!/[((r)2)!]2)

Tardigrade Admin · Accepted Answer

Tm+1= nCm (x)n-m((1/x))m = nCm xn-2m n-3m = r ⇒ m = (n-r/2) Coefficient of xr = nC((n-r/2))=(n!/((n-r)2)!((n+r)2)!

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

$\frac{n !}{( r ! ) ^{2}}$

$\frac{n !}{( r + 1 )! ( r - 1 )!}$

$\frac{n !}{( \frac{n + r}{2} ) ! ( \frac{n - r}{2} ) !}$

$\frac{n !}{[ ( \frac{r}{2} ) ! ] ^{2}}$

$T_{m + 1} =^{n} C_{m} (x)^{n - m} (\frac{1}{x})^{m}$
$=^{n} C_{m} x^{n - 2 m}$
$n - 3 m = r \Rightarrow m = \frac{n - r}{2}$
Coefficient of $x^{r} =^{n} C_{(\frac{n - r}{2})} = \frac{n !}{( \frac{n - r}{2} ) ! ( \frac{n + r}{2} ) !}$

Q. If xr occurs in the expansion of (x+x1​)n, then its coefficient is

(r!)2n!​

(r+1)!(r−1)!n!​

(2n+r​)!(2n−r​)!n!​

[(2r​)!]2n!​