Find a, if the coefficients of

Question:

Find a, if the coefficients of x2 and x3 in the expansion of (3 + ax)9 are equal.

Solution:

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

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

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

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

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

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

$=\frac{9}{7}$

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