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Question
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
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