Mathematics - Binomial Theorem Question with Solution | TestHub

We have, $\left(1+ax+b{x}^2\right){\left(1-2x\right)}^{18}=a_0+a_{1}x+a_2{x}^2+.......$
$\text{=}\left(1+\mathrm{ax}+{\mathrm{bx}}^2\right)\left(1-{}^{18}{\mathrm{C}}_1·2\mathrm{x}+{}^{18}{\mathrm{C}}_2·2^2{\mathrm{x}}^2+.....{}^{18}{C}_{18}{\left(2\mathrm{x}\right)}^{18}\right)$

$\Rightarrow {\mathrm{a}}_3={{}^{-18}\mathrm{C}}_3\times 8+{{}^{18}\mathrm{C}}_2\times 4\mathrm{a}-36\mathrm{b}$

$\Rightarrow {\mathrm{a}}_4={{}^{18}\mathrm{C}}_4\times 16-{{}^{18}\mathrm{C}}_3\times 8\mathrm{a}+{{}^{18}\mathrm{C}}_2\times 4\times \mathrm{b}$

$102a-6b=1088 ...\left(i\right)$

$64a-6b=480 ...\left(ii\right)$

Solution of the above two equation gives.

$a=16$
$\text{b}=\frac{272}{3}$