Find a if the 17^{text{th}} and 18^{text{th}} | vedclass

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

Question

(A) The $(r+1)^{	ext{th}}$ term of the expansion $(x+y)^n$ is given by $T_{r+1} = {^nC_r} x^{n-r} y^r$.For the $17^{	ext{th}}$ term,we have $r+1 = 1

Vedclass · Accepted Answer

$1$

Vedclass · Answer

(A) The $(r+1)^{	ext{th}}$ term of the expansion $(x+y)^n$ is given by $T_{r+1} = {^nC_r} x^{n-r} y^r$.For the $17^{	ext{th}}$ term,we have $r+1 = 17$,i.e.,$r = 16$.Therefore,$T_{17} = T_{16+1} = {^{50}C_{16}} (2)^{50-16} a^{16} = {^{50}C_{16}} 2^{34} a^{16}$.Similarly,for the $18^{	ext{th}}$ term,$r = 17$.Therefore,$T_{18} = T_{17+1} = {^{50}C_{17}} (2)^{50-17} a^{17} = {^{50}C_{17}} 2^{33} a^{17}$.Given that $T_{17} = T_{18}$,we have:${^{50}C_{16}} 2^{34} a^{16} = {^{50}C_{17}} 2^{33} a^{17}$.Dividing both sides by ${^{50}C_{17}} 2^{33} a^{16}$,we get:$a = \frac{{^{50}C_{16}} \cdot 2^{34}}{{^{50}C_{17}} \cdot 2^{33}} = \frac{{^{50}C_{16}}}{{^{50}C_{17}}} \cdot 2$.Using the formula $\frac{{^nC_r}}{{^nC_{r-1}}} = \frac{n-r+1}{r}$,where $n=50$ and $r=17$:$\frac{{^{50}C_{16}}}{{^{50}C_{17}}} = \frac{17}{50-17+1} = \frac{17}{34} = \frac{1}{2}$.Thus,$a = \frac{1}{2} 	imes 2 = 1$.

Find $a$ if the $17^{\text{th}}$ and $18^{\text{th}}$ terms of the expansion $(2 + a)^{50}$ are equal.

Similar Questions

In the expansion of ${\left( \frac{1 + x}{1 - x} \right)^2}$,the coefficient of ${x^n}$ is:

If $\left(\frac{3^{6}}{4^{4}}\right) k$ is the term independent of $x$ in the binomial expansion of $\left(\frac{x}{4}-\frac{12}{x^{2}}\right)^{12}$,then $k$ is equal to ...... .

The ratio of the coefficient of $x^2$ to the coefficient of $x^{10}$ in the expansion of $(x^5 + 4 \cdot 3^{-\log_{\sqrt{3}}\sqrt{x^3}})^{10}$ is

$A$ possible value of $x$,for which the ninth term in the expansion of $\left\{3^{\log _{3} \sqrt{25^{x-1}+7}}+3^{\left(-\frac{1}{8}\right) \log _{3}\left(5^{x-1}+1\right)}\right\}^{10}$ in the increasing powers of $3^{\left(-\frac{1}{8}\right) \log _{3}\left(5^{x-1}+1\right)}$ is equal to $180$,is:

If $a_r$ is the coefficient of $x^{10-r}$ in the Binomial expansion of $(1+x)^{10}$,then $\sum \limits_{r=1}^{10} r^3\left(\frac{a_r}{a_{r-1}}\right)^2$ is equal to

Vedclass Test Series

Exam Paper Generator

Online Exam Module