Show that the middle term in the expansion of $(1+x)^{2 n}$ is
$\frac{1.3 .5 \ldots(2 n-1)}{n !} 2 n\, x^{n},$ where $n$ is a positive integer.
As $2 n$ is even, the middle term of the expansion $(1+x)^{2 n}$ is $\left(\frac{2 n}{2}+1\right)^{\text {th }}$
i.e., $(n+1)^{\text {th }}$ term which is given by,
${T_{n + 1}} = {\,^{2n}}{C_n}{(1)^{2n - n}}{(x)^n} = {\,^{2n}}{C_n}{x^n} = \frac{{(2n)!}}{{n!n!}}{x^n}$
$=\frac{2 n(2 n-1)(2 n-2) \ldots 4.3 .2 .1}{n ! n !} x^{n}$
$=\frac{1.2 .3 .4 \ldots(2 n-2)(2 n-1)(2 n)}{n ! n !} x^{n}$
$=\frac{[1.3 .5 \ldots(2 n-1)][2.4 .6 \ldots .(2 n)]}{n ! n !} x^{n}$
$=\frac{[1.3 .5 \ldots(2 n-1)] 2^{n}[1.2 .3 \dots n]}{n ! n !} x^{n}$
$=\frac{[1.3 .5 \ldots(2 n-1)] n !}{n ! n !} 2^{n} \cdot x^{n}$
$=\frac{1.3 .5 \ldots(2 n-1)}{n !} 2^{n} x^{n}$
The number of terms in the expansion of ${\left( {\sqrt[4]{9} + \sqrt[6]{8}} \right)^{500}}$, which are integers is
The term independent of $x$ in the expansion of $\left[\frac{x+1}{x^{2 / 3}-x^{1 / 3}+1}-\frac{x-1}{x-x^{1 / 2}}\right]^{10}, x \neq 1,$ is equal to ....... .
The coefficient of ${x^{53}}$ in the following expansion $\sum\limits_{m = 0}^{100} {{\,^{100}}{C_m}{{(x - 3)}^{100 - m}}} {.2^m}$is
The greatest coefficient in the expansion of ${(1 + x)^{2n + 2}}$ is
If for some positive integer $n,$ the coefficients of three consecutive terms in the binomial expansion of $(1+x)^{n+5}$ are in the ratio $5: 10: 14,$ then the largest coefficient in this expansion is