Find the sum of the coefficients of two middle terms in the binomial expansion of \[\left( 1 + x \right)^{2n - 1}\]

#### Solution

\[\left( 1 + x \right)^{2n - 1} \]

\[\text{ Here, n is an odd number .} \]

\[\text{ Therefore, the middle terms are } \left( \frac{2n - 1 + 1}{2} \right)^{th} \text{ and } \left( \frac{2n - 1 + 1}{2} + 1 \right)^{th} , i . e . , n^{th} \text{ and } (n + 1 )^{th} \text{ terms } . \]

\[\text{ Now, we have} \]

\[ T_n = T_{n - 1 + 1} \]

\[ =^{2n - 1}{}{C}_{n - 1} \left( x \right)^{n - 1} \]

\[\text{ And } , \]

\[ T_{n + 1} = T_{n + 1} \]

\[ = ^{2n - 1}{}{C}_n \left( x \right)^n \]

\[ \therefore \text{ the coefficients of two middle terms are } ^{2n - 1}{}{C}_{n - 1} \text{ and } ^{2n - 1}{}{C}_n . \]

\[Now, \]

\[^{2n - 1} C_{n - 1} +^{2n - 1} C_n =^{2n} C_n\]

Hence, the sum of the coefficients of two middle terms in the binomial expansion of