Question

The middle term in the expansion of $(1 + x)^{2n}$ is

• A. $\frac{1\cdot 3\cdot 5....(2n-1)}{n} x^n$
• B. $\frac{1\cdot 3\cdot 5....(2n-1)}{n!} 2^{n-1} x^n$
• C. $\frac{1\cdot 3\cdot 5....(2n-1)}{n!} x^n$
• D. $\frac{1\cdot 3\cdot 5....(2n-1)}{n!} 2^{n} x^n$

Answer 1

Answer: (D)

Total term = $(2n + 1)$
Middle term = $\frac{(2n+1) +1}{2} = (n +1)^{th}$ term
$T_{n+1} =\,{}^{2n}C_{n} x^{n} $
$=\frac{2n!}{n!n!} x^{n}$
$ =\frac{2^{n} n!\left\{1\cdot 3\cdot 5..... \left(2n-1\right)\right\}}{n!n!}x^{n} $
$=\left\{\frac{1\cdot 3\cdot 5 .....\left(2n-1\right)}{n!}\right\}2^{n }x^{n}$