Question

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

Answer 1

We have,

$(3+ax{)}^9={}^9{C}_0\times 3^9+{}^9{C}_1\times 3^8\times (ax{)}^1+{}^9{C}_2\times 3^7\times (ax{)}^2+{}^9{C}_3\times 3^6\times (ax{)}^3+...$

$\because$ Coefficient of $x^2={}^9{C}_2\times 3^7\times a^2$ and, Coefficient of $x^3={}^9{C}_3\times 3^6\times a^3$

Now, Coefficient of $x^2$ =Coefficient of $x^3$

$\Rightarrow {}^9{C}_2\times 3^7\times a^2={}^9{C}_3\times 3^6\times a^3$

$\Rightarrow 36\times 3^7\times a^2=84\times 3^6\times a^3$

$\Rightarrow a=\frac{36\times 3^7}{84\times 3^6}=\frac{9}{7}$