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

Explanation

Given

`(3+a x)^{9}`
`=\ ^{9} C_{0} \cdot 3^{9} \cdot(a x)^{0}+\ ^{9} C_{1} \cdot 3^{8} \cdot(a x)^{1}+\ ^{9} C_{2} \cdot 3^{7} \cdot(a x)^{2}+\ ^{9} C_{3} \cdot 3^{6} \cdot(a x)^{3}+\ldots`

We have Coefficient of `x^{2}=` Coefficient of `x^{3}`

`\ ^{9} C_{2} \times 3^{7} a^{2}=\ ^{9} C_{3} \times 3^{6} a^{3}`

`\Rightarrow a=\frac{\ ^{9} C_{2}}{\ ^{9} C_{3}} \times 3`

`=\frac{9 ! \times 3 ! \times 6 ! \times 3}{2 ! \times 7 ! \times 9 !}`

`=\frac{9}{7}`