In the expansion of ${(1 + x)}^{43}$ the coefficients of the $(2 r + 1)$ th and the $(r + 2)$ th terms are equal; find the $r$ .

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Solution

Given in the expansion of
$(1 + x)^{43}$ the coefficient of $(2 r + 1)^{t h}$ and coefficient of $(r + 2)^{t h}$ are equal
$_{2 r}^{43} C =_{r + 1}^{43} C$
Either $2 r = r + 1 \Rightarrow r = 1$ or
$2 r + r + 1 = 43 \Rightarrow r = 14$
Hence $r = 1, 14$

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