In the expansion of (1+x)left(1-x^2right)left | vedclass

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

Question

$\begin{aligned} & (1+x)\left(1-x^2ight)\left(1+\frac{3}{x}+\frac{3}{x^2}+\frac{1}{x^3}ight)^5 \ & =(1+x)\left(1-x^2ight)\left(\left(1+

Vedclass · Accepted Answer

$118$

Vedclass · Answer

$\begin{aligned} & (1+x)\left(1-x^2ight)\left(1+\frac{3}{x}+\frac{3}{x^2}+\frac{1}{x^3}ight)^5 \ & =(1+x)\left(1-x^2ight)\left(\left(1+\frac{1}{x}ight)^3ight)^5 \ & =\frac{(1+x)^2(1-x)(1+x)^{15}}{x^{15}} \ & =\frac{(1+x)^{17}-x(1+x)^{17}}{x^{15}} \ & =\operatorname{coeff}\left(x^3ight) 	ext { in the expansion } \approx \operatorname{coeff}\left(x^{18}ight) 	ext { in } \ & (1+x)^{17}-x(1+x)^{17} \ & =0-1 \ & =-1 \ & 	ext { coeff }\left(x^{-13}ight) 	ext { in the expansion } \approx \operatorname{coeff}\left(x^2ight) \quad 	ext { in }\end{aligned}$

$(1+x)^{17}-x(1+x)^{17}$

$=0-1$

$=-1$

$\operatorname{coeff}\left(\mathrm{x}^{-13}ight)$ in the expansion $\approx \operatorname{coeff}\left(\mathrm{x}^2ight)$ in

$(1+x)^{17}-x(1+x)^{17}$

$=\left(\begin{array}{c}17 \ 2\end{array}ight)-\left(\begin{array}{c}17 \ 1\end{array}ight)$

$=17 	imes 8-17$

$=17 	imes 7$

$=119$

Hence Answer $=119-1=118$

In the expansion of $(1+x)\left(1-x^2\right)\left(1+\frac{3}{x}+\frac{3}{x^2}+\frac{1}{x^3}\right)^5, x \neq 0$, the sum of the coefficient of $x^3$ and $x^{-13}$ is equal to

Similar Questions

If for positive integers $r > 1,n > 2$ the coefficient of the ${(3r)^{th}}$ and ${(r + 2)^{th}}$ powers of $x$ in the expansion of ${(1 + x)^{2n}}$ are equal, then

The largest term in the expansion of ${(3 + 2x)^{50}}$ where $x = \frac{1}{5}$ is

In the expansion of ${({5^{1/2}} + {7^{1/8}})^{1024}}$, the number of integral terms is

The first $3$ terms in the expansion of ${(1 + ax)^n}$ $(n \ne 0)$ are $1, 6x$ and $16x^2$. Then the value of $a$ and $n$ are respectively

If the coefficients of $x^{7}$ and $x^{8}$ in the expansion of $\left(2+\frac{x}{3}\right)^{n}$ are equal, then the value of $n$ is equal to $.....$