Question

If the coefficient of the middle term in the expansion of $(1+x{)}^{2n+2}$ is $\alpha$ and the coefficients of middle terms in the expansion of $(1+x{)}^{2n+1}$ are $\beta$ and $\gamma$, then relation between $\alpha ,\beta$ and $\gamma$ is-

• A. $\alpha +\beta +\gamma =0$
• B. $\alpha =\beta +\gamma$
• C. $\beta =\alpha +\gamma$
• D. $\gamma =\beta +\alpha$

Answer 1

The correct option is B $\alpha =\beta +\gamma$

$(n+2{)}^{th}$ term is the middle term in the expansion of $(1+x{)}^{2n+2}$.
Therefore, $\alpha {=}^{2n+2}{C}_{n+1}.$
$(n+1{)}^{th}$ and $(n+2{)}^{th}$ terms are middle terms in the expansion of $(1+x{)}^{2n+1}$
Therefore, $\beta ={}^{2n+1}{C}_n$ and $\gamma ={}^{2n+1}{C}_{n+1}$
Also,
${}^{2n+1}{C}_n+{}^{2n+1}{C}_{n+1}={}^{2n+2}{C}_{n+1}$
$(\because {}^n{C}_r+{}^n{C}_{r+1}={}^{n+1}{C}_{r+1})$
$\Rightarrow \beta +\gamma =\alpha$