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General Term of Binomial Expansion

If in the exp...

Question

If in the expansion of $(1 + x)^{43}$ , the coefficient of $(2 r + 1) t h$ term is equal to coefficient of $(r + 2)^{t h}$ term. Find r ??

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Solution

$(1 + x)^{43} =^{43} C_{0} +^{43} C_{1} +^{43} C_{2} x^{2}^{42} C_{n} x^{n}$
$r^{t h} t e r m =^{43} C (r - 1) x^{(r - 1)}$
coefficient of $(2 r + 1)^{t h} t e r m =^{43} C_{2 r} . . . . . . (1)$
coefficient of $(r + 2)^{t h} t e r m =^{43} C_{(r + 1)} . . . . . . . (2)$
Given $(1) = (2)$
$∴^{43} C_{2 r} =^{43} C_{(r + 1)}$
$\frac{43!}{(43 - 2 r)! (24)!} = \frac{43!}{(43 - r - 1)! (r + 1)!}$
$\Rightarrow 43 = (2 r + r +!)$ $[(^{n} C_{r} =^{n} C_{p}) \Rightarrow r + p = n]$
$\Rightarrow 42 = 3 r$
$\Rightarrow r = 14$

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