The coefficient of 1 / x in the expansion of (1+ x ) n (1+ 1 / x ) is

Question

The coefficient of 1 / x in the expansion of (1+ x ) n (1+ 1 / x ) is (A) ( n !/( n -1) !( n +1) !) (B) (2 n !/(n-1) !(n+1) !) (C) ((2 n) !/(2 n-1) !(2 n+1) !) (D) none of these

Tardigrade Admin · Accepted Answer

Coefficient of x -1 in (1+ x ) n (1+(1/ x )) n = Coefficient of x -1 in ((1+ x )2 n / x n ) = Coefficient of x n -1 in (1+ x )2 n = 2 n C n -1=((2 n ) !/( n -1) !( n +1) !)

Q. The coefficient of $1/ x$ in the expansion of $(1 + x)^{n} (1 +$ $1/ x)$ is

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

$\frac{2 n !}{( n - 1 )! ( n + 1 )!}$

$\frac{( 2 n )!}{( 2 n - 1 )! ( 2 n + 1 )!}$

none of these

Coefficient of $x^{- 1}$ in $(1 + x)^{n} (1 + \frac{1}{x})^{n}$
$=$ Coefficient of $x^{- 1}$ in $\frac{( 1 + x ) ^{2 n}}{x ^{n}}$
$=$ Coefficient of $x^{n - 1}$ in $(1 + x)^{2 n}$
$=^{2 n} C_{n - 1} = \frac{( 2 n )!}{( n - 1 )! ( n + 1 )!}$

Q. The coefficient of 1/x in the expansion of (1+x)n(1+ 1/x) is

(n−1)!(n+1)!n!​

(n−1)!(n+1)!2n!​

(2n−1)!(2n+1)!(2n)!​