# True or False : The sum of coefficients of the two middle terms in the expansion of $(1+x)^{2n-1}$ is equal to $2n-1C_n$.

$\begin{array}{1 1}(A)\;\text{True}\\(B)\;\text{False}\end{array}$

Toolbox:
• When n is odd,middle term $\Rightarrow (\large\frac{n+1}{2})^{th}$ and $(\large\frac{n+3}{2})^{th}$
$(1+x)^{2n-1}$
Middle term $\Rightarrow \big(\large\frac{2n-1+1}{2}\big)^{th}$,$\big(\large\frac{2n-1+3}{2}\big)^{th}$
(i.e) $n^{th}$ and $(n+1)^{th}$ terms
$T_n=T_{(n-1)+1}=2n-1C_{n-1}(1)^{(2n-1)-(n-1)}x^{n-1}$
$\Rightarrow 2n-1 C_{n-1} x^{n-1}$
$T_{n+1}=2n-1C_n(1) ^{(2n-1)-n}x^n$
$\Rightarrow 2n-1C_nx^n$
So the coefficient of two middle term in the expansion of $(1+x)^{2n-1}$ are $2n-1C_{n-1}$ and $2n-1C_n$
Sum of these coefficients
$\Rightarrow 2n-1C_{n-1}+2n-1C_n x^n$
$nC_{r-1}+nC_r=n+1C_r$
$\Rightarrow (2n-1+1C_n)$
$\Rightarrow 2nC_n$
Hence the given statement is False.
answered Jun 25, 2014