Question

If A is the coefficient of the middle term in the expansion of $(1+x{)}^{2n}$ and B and C are the coefficients of two middle terms in the expansion of $(1+x{)}^{2n-1}$, then

• A. $A+B=C$
• B. $B+C=A$
• C. $C+A=B$
• D. $A+B+C=0$

Answer 1

Answer: (B)

$B+C=A$

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