If the coefficients of the (2r+1)^{text{th}} | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) The general term in the expansion of $(1+x)^n$ is given by $T_{k+1} = {^nC_k} x^k$. For the expansion of $(1+x)^{42}$,the coefficients are: Coeffi

Vedclass · Accepted Answer

$14$

Vedclass · Answer

(B) The general term in the expansion of $(1+x)^n$ is given by $T_{k+1} = {^nC_k} x^k$. For the expansion of $(1+x)^{42}$,the coefficients are: Coefficient of $(2r+1)^{	ext{th}}$ term is ${^{42}C_{2r}}$. Coefficient of $(r+1)^{	ext{th}}$ term is ${^{42}C_r}$. Given that these coefficients are equal: ${^{42}C_{2r}} = {^{42}C_r}$. Using the property ${^nC_x} = {^nC_y}$,we have two cases: Case $1$: $x = y$ $\Rightarrow 2r = r$ $\Rightarrow r = 0$. Case $2$: $x + y = n$ $\Rightarrow 2r + r = 42$ $\Rightarrow 3r = 42$ $\Rightarrow r = 14$. Since $r$ must be a positive integer in this context,$r = 14$ is the valid solution.

If the coefficients of the $(2r+1)^{\text{th}}$ term and the $(r+1)^{\text{th}}$ term in the expansion of $(1+x)^{42}$ are equal,then $r$ can be

Similar Questions

The term independent of $x$ in the expansion of $\left( 9x - \frac{1}{3\sqrt{x}} \right)^{18}, x > 0$,is $\alpha$ times the corresponding binomial coefficient. Then $\alpha$ is:

The sum of the real values of $x$ for which the middle term in the binomial expansion of ${\left( {\frac{{{x^3}}}{3} + \frac{3}{x}} \right)^8}$ equals $5670$ is

If the constant term in the binomial expansion of $\left(\frac{x^{5/2}}{2} - \frac{4}{x^{\ell}}\right)^9$ is $-84$ and the coefficient of $x^{-3\ell}$ is $2^{\alpha}\beta$,where $\beta < 0$ is an odd number,then $|\alpha\ell - \beta|$ is equal to

The sum of the rational terms in the binomial expansion of $(2^{1/2} + 3^{1/5})^{10}$ is

In the expansion of ${\left( 2{x^2} - \frac{1}{x} \right)^{12}}$,the term independent of $x$ is: (in $^{th}$)

Vedclass Test Series

Exam Paper Generator

Online Exam Module