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 question, we need to analyze both the assertion and the reason provided in the context of wave optics, specifically interference.

### Step-by-Step Solution:

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

2. **Path Difference and Phase Difference**:
- The path difference \( \Delta x \) is given as \( \frac{\lambda}{3} \).
- The phase difference \( \phi \) can be calculated using the formula:
\[ \phi = \frac{2\pi}{\lambda} \Delta x \]
- Substituting \( \Delta x = \frac{\lambda}{3} \):
\[ \phi = \frac{2\pi}{\lambda} \cdot \frac{\lambda}{3} = \frac{2\pi}{3} \]

3. **Calculating Resultant Intensity**:
- The resultant intensity \( I \) for two coherent sources can be expressed as:
\[ I = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos(\phi) \]
- Since both sources have equal intensity \( I_1 = I_2 = I_0 \):
\[ I = I_0 + I_0 + 2\sqrt{I_0 I_0} \cos\left(\frac{2\pi}{3}\right) \]
- Simplifying further:
\[ I = 2I_0 + 2I_0 \cos\left(\frac{2\pi}{3}\right) \]
- The value of \( \cos\left(\frac{2\pi}{3}\right) \) is \( -\frac{1}{2} \):
\[ I = 2I_0 - I_0 = I_0 \]

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

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.
- This is generally true, but it is important to note that the resultant intensity is not simply the average of the two intensities; it also depends on the phase difference between the waves.

6. **Conclusion for Reason**:
- The reason is incorrect as stated. The resultant intensity is not merely the average of the two individual intensities; it also includes the interference term that depends on the phase difference.

### Final Conclusion:
- The assertion is correct, but the reason is not correctly stated. Therefore, the correct answer is that the assertion is true, but the reason is false.