Question

if the coefficient of the middle term in the expansion of$(1+x{)}^{2n+2}$and $p$ 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)

$p=q+r$

$(1+x{)}^{2n+2}$.
The middle term will be $(\frac{N}{2}+1)$ th term.
$=\frac{2n+2}{2}+1$ th term
$=(n+2{)}^{th}$ term
Hence coefficient of the middle term will be
${}^{2n+2}{C}_{n+1}$
$=p$
For
$(1+x{)}^{2n+1}$.
The middle terms will be $(\frac{N+1}{2}+1)$ and $\left(\frac{N+1}{2}\right)$ terms
$=\frac{2n+2}{2}+1$ th term and ${\frac{2n+2}{2}}^{th}$ term.
$=(n+2{)}^{th}$ term and $(n+1{)}^{th}$ term.
Hence coefficient of the middle terms will be
${}^{2n+1}{C}_{n+1}=q$ and ${}^{2n+1}{C}_n=r$
By the properties of binomial coefficients.
${}^{2n+1}{C}_{n+1}+{}^{2n+1}{C}_n$
$={}^{2n+2}{C}_{n+1}$
Hence $q+r=p$