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

Open in App

Solution

Here $(3 + a x)^{9}$

= $9_{C_{0}} (3)^{9} + 9_{C_{1}} (3)^{8} (a x) + 9_{C_{2}} (3)^{7} (a x)^{2} + 9_{C_{3}} (3)^{6} (a x)^{3} + . . .$

= $9_{C_{0}} (3)^{9} + 9_{C_{1}} (3)^{8} . a . x + 9_{C_{2}} (3)^{7} (a)^{2} . x^{2} + 9_{C_{3}} (3)^{6} . a^{3} x^{3} + . . .$

$∴$ 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 $9_{C_{2}} (3)^{7} a^{2} = 9_{C_{3}} (3)^{6} a^{3}$

$\Rightarrow 36. 3^{7} a^{2} = 84. 3^{6} . a^{3}$

$\Rightarrow a = \frac{36. 3^{7}}{{84.3}^{6}} = \frac{108}{84} = \frac{9}{7}$ .

Suggest Corrections

Similar questions

a

x^{2}

x^{3}

{(3 + a x)}^{9}

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

(3 + a x)^{9}

x^{2}

x^{3}

a =

| x | < 1

x^{n}

{(1 + x + x^{2} + x^{3} + . . . . . .)}^{2}

x^{2}

x^{3}

{(3 + a x)}^{9}

^{'} a^{'}

Join BYJU'S Learning Program