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Question
If the coefficients of (2r + 1)th term and (r + 2)th term in the expansion of (1 + x)43 are equal, find r.
Solution
\[\text{ Given } : - (1 + x )^{43} \]
\[\text{ We know that the coefficient of the rth term in the expansion of } (1 + x )^n \text{ is} ^{n}{}{C}_{r - 1} \]
\[\text{ Therefore, the coefficients of the (2r + 1)th and (r + 2)th terms in the given expression are }^{43}{}{C}_{2r + 1 - 1} \text{ and } ^{43}{}{C}_{r + 2 - 1} \]
\[\text{ For these coefficients to be equal, we must have:} \]
\[ \Rightarrow 2r = r + 1 \text{ or } , 2r + r + 1 = 43 [ \because ^{n}{}{C}_r = ^{n}{}{C}_s \Rightarrow r = s \text{ or } r + s = n]\]
\[ \Rightarrow r = 14 \left[ \because \text{ for r = 1 it gives the same term } \right]\]
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