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 x^{2} 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 x^{2} and in \( (T_{r+1})\), we obtain r = 2

Thus, the coefficient of x^{2} is

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

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

Assuming that x^{3} 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 x^{3} and in \( (T_{r+1})\), we obtain k =3

Thus, the coefficient of x^{3} 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 x^{2} and x^{3} 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 \( \dfrac{9}{7}\)

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