If the coefficents of {x^3} and {x^4} in | vedclass

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

Question

In the expansion of $\left(1+a x+b x^{2}\right)(1-2 x)^{18},$

Coefficient of $x^{3}$ in $\left(1+a x+b x^{2}\right)(1-2 x)^{18}$

$=$ Coefficient

Vedclass · Accepted Answer

($16$,$\frac{{272}}{3}$)

Vedclass · Answer

In the expansion of $\left(1+a x+b x^{2}ight)(1-2 x)^{18},$

Coefficient of $x^{3}$ in $\left(1+a x+b x^{2}ight)(1-2 x)^{18}$

$=$ Coefficient of $x^{3}$ in $(1-2 x)^{18}$

${+	ext { Coefficient of } x^{2} 	ext { in a }(1-2 x)^{	ext {18 }}}$

${	ext { + Coefficient of } x 	ext { in } b(1-2 x)^{	ext {18 }}}$

$=-^{18} \mathrm{C}_{3} \cdot 2^{3}+\mathrm{a}^{18} \mathrm{C}_{2} \cdot 2^{2}-\mathrm{b}^{18} \mathrm{C}_{1} \cdot 2$

$\because-^{18} C_{3} \cdot 2^{3}+a^{18} C_{2} \cdot 2^{2}-b^{18} C_{1} \cdot 2=0$

$\Rightarrow \frac{18 	imes 17 	imes 16}{3 	imes 2} \cdot 8+a \cdot \frac{18 	imes 17}{2} 2^{2}-b \cdot 18 \cdot 2=0$

$\Rightarrow 17 a-b=\frac{34 	imes 16}{3}..........(i)$

Similarly, coefficient of $x^{4}$

$^{18} \mathrm{C}_{4} \cdot 2^{4}-\mathrm{a} \cdot^{18} \mathrm{C}_{3} 2^{3}+b \cdot^{18} \mathrm{C}_{2} \cdot 2^{2}=0$

$32 a-32 b=240..........(ii)$

On solving Eqs. $(i)$ and $(ii)$, we get

$a=16 	ext { and } b=\frac{272}{3}$

If the coefficents of ${x^3}$ and ${x^4}$ in the expansion of $\left( {1 + ax + b{x^2}} \right){\left( {1 - 2x} \right)^{18}}$ in powers of $x$ are both zero, then $ (a,b) $ is equal to

Similar Questions

The coefficient of $x^4$ in ${\left[ {\frac{x}{2}\,\, - \,\,\frac{3}{{{x^2}}}} \right]^{10}}$ is :

If ${\left( {2 + \frac{x}{3}} \right)^{55}}$ is expanded in the ascending powers of $x$ and the coefficients of powers of $x$ in two consecutive terms of the expansion are equal, then these terms are

The middle term in the expansion of ${(1 + x)^{2n}}$ is

Find the cocfficient of $x^{5}$ in $(x+3)^{8}$

$x^r$ occurs in the expansion of ${\left( {{x^3} + \frac{1}{{{x^4}}}} \right)^n}$ provided -