Question

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

• A. ${}^{2n}{C}_n$
• B. ${}^{2n}{C}_{n+1}{x}^{n+1}$
• C. ${}^{2n}{C}_{n-1}{x}^{n-1}$
• D. $\frac{\mathrm{1.3.5...}(2n-1)}{n!}.2^n{x}^n$

Answer 1

Answer: (A)

Middle term in $(1+x{)}^{2n}=T_{n+1}$
$={}^{2n}{c}_n{x}^n=\frac{2n!}{n!n!}$
$=\frac{\left[\mathrm{1.3.5......}\left(2n-1\right)\right]\left[\mathrm{2.4.6...2}n\right]}{n!n!}$
$=\frac{\left[\mathrm{1.3.5.......}\left(2n-1\right)\right]2^n}{n!}{x}^n$