Question

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

Answer 1

Here $(3+ax{)}^9$

=$9_{C_0}(3{)}^9+9_{C_1}(3{)}^8\left(ax\right)+9_{C_2}\left(3{)}^7\right(ax{)}^2+9_{C_3}\left(3{)}^6\right(ax{)}^3+...$

=$9_{C_0}(3{)}^9+9_{C_1}(3{)}^8.a.x+9_{C_2}\left(3{)}^7\right(a{)}^2.x^2+9_{C_3}(3{)}^6.a^3{x}^3+...$

$\therefore$ Coefficient of $x^2=9_{C_2}(3{)}^7{a}^2$

Coefficient of $x^3=9_{C_3}(3{)}^6{a}^3$

It is given that $9_{C_2}(3{)}^7{a}^2=9_{C_3}(3{)}^6{a}^3$

$\Rightarrow 36.3^7{a}^2=84.3^6.a^3$

$\Rightarrow a=\frac{36.3^7}{{84.3}^6}=\frac{108}{84}=\frac{9}{7}$.