Question

Assertion (A) : The sum of the coefficients of the middle terms in the expansion of
$(1+x{)}^{2n-1}$ is equal to ${}^{2n}{C}_n$.
Reason (R) : To find the sum of the coefficients of the two middle terms in the expansion
$(1+x{)}^{2n-1}$ the value of $n$ is any natural number.

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true, but R is not the correct explanation of A
• C. A is true, but R is false
• D. A is false, but R is true

Answer 1

The correct option is B Both A and R are true, but R is not the correct explanation of A

$(1+x{)}^{2n-1}$
Therefore the total number of terms will be $2n$ terms.
Hence there will be two middle terms since the total number of terms is even.
The middle terms will be $2n/2$th and $(2n/2+1)$th terms.
Therefore
$T_n+T_{n+1}$
$={}^{2n-1}{C}_{n-1}+{}^{2n-1}{C}_n$
$={}^{2n}{C}_n$ ...(from properties of binomial coefficient)
Hence assertion is true but reason is not the correct explanation of A.