Assertion Two coherent sources transmit waves of equal intensity `I_(0)` Resultant intensity at a point where path difference is `lambda/3` is also `I_(0)`.
Reason In interference resultant intensity at any point is the average intensity of two individual intensities.

Explanation

To solve the problem, we need to analyze both the assertion and the reason provided in the question.

### Step-by-Step Solution:

1. **Understanding the Assertion**:
- The assertion states that two coherent sources transmit waves of equal intensity \( I_0 \) and that the resultant intensity at a point where the path difference is \( \frac{\lambda}{3} \) is also \( I_0 \).

2. **Calculating Phase Difference**:
- The phase difference \( \phi \) corresponding to a path difference \( \Delta x \) is given by:
\[ \phi = \frac{2\pi}{\lambda} \Delta x \]
- Substituting \( \Delta x = \frac{\lambda}{3} \):
\[ \phi = \frac{2\pi}{\lambda} \cdot \frac{\lambda}{3} = \frac{2\pi}{3} \]

3. **Using the Intensity Formula**:
- The resultant intensity \( I \) for two coherent sources can be calculated using the formula:
\[ I = I_1 + I_2 + 2\sqrt{I_1 I_2 \cos(\phi)} \]
- Since both sources have equal intensity \( I_0 \):
\[ I = I_0 + I_0 + 2\sqrt{I_0 I_0 \cos\left(\frac{2\pi}{3}\right)} \]
- Simplifying this:
\[ I = 2I_0 + 2I_0 \cos\left(\frac{2\pi}{3}\right) \]
- The value of \( \cos\left(\frac{2\pi}{3}\right) \) is \( -\frac{1}{2} \):
\[ I = 2I_0 - 2I_0 \cdot \frac{1}{2} = 2I_0 - I_0 = I_0 \]

4. **Conclusion for Assertion**:
- The resultant intensity at the point where the path difference is \( \frac{\lambda}{3} \) is indeed \( I_0 \). Thus, the assertion is **correct**.

5. **Understanding the Reason**:
- The reason states that in interference, the resultant intensity at any point is the average intensity of the two individual intensities.
- The formula for resultant intensity \( I \) is:
\[ I = I_1 + I_2 + 2\sqrt{I_1 I_2 \cos(\phi)} \]
- The average intensity \( \bar{I} \) of two sources is given by:
\[ \bar{I} = \frac{I_1 + I_2}{2} \]
- However, the resultant intensity is not simply the average of the two intensities due to the interference term \( 2\sqrt{I_1 I_2 \cos(\phi)} \). Therefore, the reason is **incorrect**.

### Final Conclusion:
- The assertion is **correct** while the reason is **incorrect**.