Question

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

• A. $\beta -\gamma =\alpha$
• B. $\gamma -\beta =\alpha$
• C. $\beta +\gamma =\alpha$
• D. none of these

Answer 1

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

$(1+x{)}^{2n+2}$.
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
${}^{2n+2}{C}_{n+1}$
$=\alpha$.
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
${}^{2n+1}{C}_{n+1}=\beta$ and ${}^{2n+1}{C}_n=\gamma$
By the properties of binomial coefficients.
${}^{2n+1}{C}_{n+1}+{}^{2n+1}{C}_n$
$={}^{2n+2}{C}_{n+1}$
Hence $\beta +\gamma =\alpha$