If the coefficient of `x^(2) " and "x^(3)` are equal in the expansion of `(3+ax)^(9)`, then find the value of 'a'

Explanation

To solve the problem, we need to find the value of 'a' such that the coefficients of \(x^2\) and \(x^3\) in the expansion of \((3 + ax)^9\) are equal.

### Step-by-Step Solution:

1. **Identify the General Term**:
The general term \(T_{r+1}\) in the expansion of \((3 + ax)^9\) is given by:
\[ T_{r+1} = \binom{9}{r} (3)^{9-r} (ax)^r \]

2. **Find the Coefficient of \(x^2\)**:
To find the coefficient of \(x^2\), we set \(r = 2\):
\[ T_{3} = \binom{9}{2} (3)^{9-2} (ax)^2 = \binom{9}{2} (3)^7 a^2 \]
The coefficient of \(x^2\) is:
\[ C_2 = \binom{9}{2} \cdot 3^7 \cdot a^2 \]

3. **Find the Coefficient of \(x^3\)**:
To find the coefficient of \(x^3\), we set \(r = 3\):
\[ T_{4} = \binom{9}{3} (3)^{9-3} (ax)^3 = \binom{9}{3} (3)^6 (ax)^3 \]
The coefficient of \(x^3\) is:
\[ C_3 = \binom{9}{3} \cdot 3^6 \cdot a^3 \]

4. **Set the Coefficients Equal**:
According to the problem, the coefficients of \(x^2\) and \(x^3\) are equal:
\[ \binom{9}{2} \cdot 3^7 \cdot a^2 = \binom{9}{3} \cdot 3^6 \cdot a^3 \]

5. **Simplify the Equation**:
Dividing both sides by \(3^6\) gives:
\[ \binom{9}{2} \cdot 3 \cdot a^2 = \binom{9}{3} \cdot a^3 \]
Rearranging gives:
\[ \binom{9}{2} \cdot 3 = \binom{9}{3} \cdot a \]

6. **Calculate the Binomial Coefficients**:
Calculate \(\binom{9}{2}\) and \(\binom{9}{3}\):
\[ \binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36 \]
\[ \binom{9}{3} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \]

7. **Substitute the Values**:
Substitute the values into the equation:
\[ 36 \cdot 3 = 84 \cdot a \]
Simplifying gives:
\[ 108 = 84a \]

8. **Solve for 'a'**:
\[ a = \frac{108}{84} = \frac{9}{7} \]

### Final Answer:
The value of \(a\) is \(\frac{9}{7}\).