Question

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

 

Answer 1

\[\left( 1 + x \right)^{2n - 1} \]
\[\text{ Here, n is an odd number .}  \]
\[\text{ Therefore, the middle terms are } \left( \frac{2n - 1 + 1}{2} \right)^{th} \text{ and }  \left( \frac{2n - 1 + 1}{2} + 1 \right)^{th} , i . e . , n^{th} \text{ and }  (n + 1 )^{th} \text{ terms }  . \]
\[\text{ Now, we have} \]
\[ T_n = T_{n - 1 + 1} \]
\[ =^{2n - 1}{}{C}_{n - 1} \left( x \right)^{n - 1} \]
\[\text{ And } , \]
\[ T_{n + 1} = T_{n + 1} \]
\[ = ^{2n - 1}{}{C}_n \left( x \right)^n \]
\[ \therefore \text{ the coefficients of two middle terms are } ^{2n - 1}{}{C}_{n - 1} \text{ and } ^{2n - 1}{}{C}_n . \]
\[Now, \]
\[^{2n - 1} C_{n - 1} +^{2n - 1} C_n =^{2n} C_n\]

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

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