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Question
Prove that the coefficient of (r + 1)th term in the expansion of (1 + x)n + 1 is equal to the sum of the coefficients of rth and (r + 1)th terms in the expansion of (1 + x)n.
Solution
\[\text{ Coefficient of the (r + 1)th term in } (1 + x )^{n + 1}\text{ is } ^{n + 1}{}{C}_r \]
\[\text{ Sum of the coefficients of the rth and (r + 1)th terms in } (1 + x )^n = ^ {n}{}{C}_{r - 1} +^{n}{}{C}_r \]
\[ =^{n + 1}{}{C}_r \left[ \because ^{n}{}{C}_{r + 1} + ^{n}{}{C}_r =^ {n + 1}{}{C}_{r + 1} \right] \]
\[\text{ Hence proved } .\]
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