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

The correct option is C

p = q + r

Since $(n+2{)}^{th}$ term is the middle term in the

expansion of $(1+x{)}^{2n+2}$, therefore p = ${}^{2n+2}{C}_{n+1}$.

Since $(n+1{)}^{th}$ and $(n+2{)}^{th}$ terms are middle terms in the expansion of $(1+x{)}^{2n+1}$,

therefore q = ${}^{2n+1}{C}_n$ and r =${}^{2n+1}{C}_{2n+1}$.

But ,${}^{2n+1}{C}_n$ +${}^{2n+1}{C}_{n+1}$ = ${}^{2n+2}{C}_{n+1}$

∴ q + r = p