Question

Find $a$ if the coefficients of $x^2$ and $x^3$ in the expansion of ${(3+ax)}^9$ are equal.

Answer 1

In the expansion of ${(a+b)}^n$ the ${(r+1)}^{th}$ term is

$T_{r+1}={n_C}_r{a}^{n-r}{b}^r$

General term in ${(3+ax)}^9$ is

$T_{r+1}={9_C}_r{3}^{9-r}{\left(ax\right)}^r={9_C}_r{3}^{9-r}{a}^r{x}^r$

Given that coefficient of $x^2=$co-efficient of $x^3$

$\Rightarrow {9_C}_2{3}^{9-2}{a}^2={9_C}_3{3}^{9-3}{a}^3$

$\Rightarrow a=\frac{{9_C}_2{3}^{9-2}}{{9_C}_3{3}^{9-3}}$

$\Rightarrow a=\frac{{9_C}_2{3}^7}{{9_C}_3{3}^6}$

$\Rightarrow a=\frac{\frac{9\times 8}{2\times 1}}{\frac{9\times 8\times 7}{3\times 2\times 1}}{3}^{7-6}=\frac{3\times 3}{7}=\frac{9}{7}$