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

It is known that (r+1)^th term \( (T_{r+1})\) in the binomial expansion of (a+b)ⁿ is given by

\( (T_{r+1})=\;^nC_ra^{n-r}b^r\) Assuming that x² occurs in the (r+1)^th term in the expansion of \( (3+ax)^9\) we obtain

\( (T_{r+1})=\;^9C_r(3)^{9-r}(ax)^r\)

= \(^9C_r(3)^{9-r}a^rx^r\)

Comparing the indices of x in x² and in \( (T_{r+1})\), we obtain r = 2

Thus, the coefficient of x² is

\( ^9C_r(3)^{9-2}a^2\)

\(= \dfrac{9!}{2!7!}(3)^7a^2=36(3)^7a^2\)

Assuming that x³ occurs in the (k+1)th term in the expansion of \( (3+ax)^9\),we obtain

\( (T_{k+1})=\;^9C_r(3)^{9-k}(ax)^k\)

= \(^9C_r(3)^{9-k}a^kx^k\)

Comparing the indices of x in x³ and in \( (T_{r+1})\), we obtain k =3

Thus, the coefficient of x³ is

\( ^9C_r(3)^{9-3}a^3\)

\( \dfrac{9!}{3!6!}(3)^6a^3=84(3)^6a^3\)

It is given that the coefficient of x² and x³ are the same.

\( 84(3)^6=36\times 3\)

\( a=\dfrac{36\times 3}{84}=\dfrac{104}{84}\)

\( a=\dfrac{9}{7}\)

Thus, the required value of a is \( \dfrac{9}{7}\)