Question

If the coefficient of the middle term in the expansion of $(1+x{)}^{2n+2}$ is p and the coefficients of middle terms in the expansion of $(1+x{)}^{2n+1}$ are $q$ and $r$, then

• A. $p+q=r$
• B. $p+r=q$
• C. $p=q+r$
• D. $p+q+r=0$

Answer 1

Answer: (C)

$p=q+r$

Middle term in the expansion of $(1+x{)}^{2n+2}$ is $(n+2{)}^{th}$ term.
So, $T_{n+2}{=}^{2n+2}{C}_{n+1}{x}^{n+1}$
$\Rightarrow p{=}^{2n+2}{C}_{n+1}$
Middle terms in the expansion of $(1+x{)}^{2n+1}$ are $(n+2{)}^{th}$ and $(n+2{)}^{th}$ term.
So, $T_{n+1}{=}^{2n+1}{C}_n{x}^n$
$T_{n+2}{=}^{2n+1}{C}_{n+1}{x}^{n+1}$
$\Rightarrow q{=}^{2n+1}{C}_n$ and $r{=}^{2n+1}{C}_{n+1}$
Now, ${}^{2n+1}{C}_n{+}^{2n+1}{C}_{n+1}{=}^{2n+2}{C}_{n+1}$ (${\because }^n{C}_r{+}^n{C}_{r-1}{=}^{n+1}{C}_r$)
$\Rightarrow q+r=p$