In the expansion of (1 + x)^{43},if the coeff | vedclass

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(C) The general term in the expansion of $(1 + x)^n$ is given by $T_{k+1} = {^nC_k} x^k$.For the $(2r + 1)^{th}$ term,$k = 2r$,so the coefficient is $

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$14$

Vedclass · Answer

(C) The general term in the expansion of $(1 + x)^n$ is given by $T_{k+1} = {^nC_k} x^k$.For the $(2r + 1)^{th}$ term,$k = 2r$,so the coefficient is ${^{43}C_{2r}}$.For the $(r + 2)^{th}$ term,$k = r + 1$,so the coefficient is ${^{43}C_{r+1}}$.Given that the coefficients are equal,we have ${^{43}C_{2r}} = {^{43}C_{r+1}}$.Using the property ${^nC_a} = {^nC_b}$,either $a = b$ or $a + b = n$.Case $1$: $2r = r + 1 \Rightarrow r = 1$.Case $2$: $2r + (r + 1) = 43$ $\Rightarrow 3r = 42$ $\Rightarrow r = 14$.Since $r = 14$ is one of the options,the correct value is $14$.

In the expansion of $(1 + x)^{43}$,if the coefficients of the $(2r + 1)^{th}$ and the $(r + 2)^{th}$ terms are equal,the value of $r$ is:

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