In the expansion of (1+x)^ 43 , if the coefficients of (2 r+1)^ th and (r+2)^ th terms are equal, then what is the value of r(r ≠ 1) ?

Correct answer: B. 14 Given, in the expansion g(1+x)^ 43 , coefficients of (2 r+1)^ th term and (r+2)^ th term are equal. Coefficient of (2 r+1)^ th term = ^ n C_ 2 r Coefficient of (r+2)^ th term =n_ C_ r+1 aligned & n_ C_ 2 r =n_ C_ r+1 & 43_ C_ 2 r =43_ C_ r+1 ( n=43) & 2 r+r+1=43 & 3 r+1=43 & 3 r+42 r=14 aligned

NDA 2018

Exam: NDA
Year: 2018
Subject: Mathematics
Chapter: Binomial Theorem

NDA 2018 · Mathematics · Binomial Theorem

In the expansion of (1+x)^ 43 , if the coefficients of (2 r+1)^ th...

In the expansion of

(1+x)^{43}

, if the coefficients of

(2 r+1)^{\text{th }}

and

(\mathrm{r}+2)^{\mathrm{th}}

terms are equal, then what is the value of

\mathrm{r}(\mathrm{r} \neq 1)

Options

Correct answer

B. 14

Step-by-step solution

Given, in the expansion

\mathrm{g}(1+\mathrm{x})^{43}

, coefficients of

(2 \mathrm{r}+1)^{\mathrm{th}}

term and

(\mathrm{r}+2)^{\mathrm{th}}

term are equal. Coefficient of

(2 \mathrm{r}+1)^{\text{th }}

term

={ }^{\mathrm{n}} \mathrm{C}_{2 \mathrm{r}}

Coefficient of

(\mathrm{r}+2)^{\text{th }}

term

=\mathrm{n}_{\mathrm{C}_{\mathrm{r}+1}}

\begin{aligned} & \mathrm{n}_{\mathrm{C}_{2 \mathrm{r}}}=\mathrm{n}_{\mathrm{C}_{\mathrm{r}+1}} \\ & \Rightarrow 43_{\mathrm{C}_{2 \mathrm{r}}}=43_{\mathrm{C}_{\mathrm{r}+1}} (\because \mathrm{n}=43)\\ & \Rightarrow 2 \mathrm{r}+\mathrm{r}+1=43 \\ & \Rightarrow 3 \mathrm{r}+1=43 \\ & \Rightarrow 3 \mathrm{r}+42 \Rightarrow \mathrm{r}=14 \end{aligned}

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