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The image you uploaded contains a mathematical problem involving geometric progression (GP). Here is the text extracted from the image:

**Problem Statement:**
- **Statement 1 (S1):** A certain progression is a geometric progression with common ratio \(\frac{2}{3}\).
- **Statement 2 (S2):** The progression is an increasing sequence with a positive common ratio.

**Question:**
Which of the above statements is/are correct?

### Solution:
To analyze this problem, let's consider the nature of a geometric progression (GP) and its common ratio:

1. **Geometric Progression (GP):** 
   - In a GP, each term after the first is obtained by multiplying the previous term by a constant called the common ratio (\(r\)).

2. **Statement 1 Analysis (S1):**
   - The common ratio is given as \(\frac{2}{3}\). Since \(\frac{2}{3}\) is positive and less than 1, this suggests that the terms of the sequence will decrease, not increase.

3. **Statement 2 Analysis (S2):**
   - This statement asserts that the sequence is increasing with a positive common ratio. However, if the common ratio is less than 1 (as it is in this case, \(\frac{2}{3}\)), the sequence cannot be increasing. The sequence would decrease because each subsequent term would be a fraction of the previous one.

**Conclusion:**
- **Statement 1 (S1):** True, because the progression is a GP with the specified common ratio.
- **Statement 2 (S2):** False, because the progression is not increasing due to the common ratio being less than 1.

Therefore, only **Statement 1** is correct.

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### Related Questions:
1. What is the general formula for the nth term of a geometric progression?
2. How does the common ratio in a GP affect whether the sequence is increasing or decreasing?
3. Can a geometric progression with a positive common ratio ever be decreasing?
4. What happens to the terms of a GP when the common ratio is exactly 1?
5. How does a geometric progression differ from an arithmetic progression?

### Tip:
Always check the sign and magnitude of the common ratio in a geometric progression to determine if the sequence is increasing, decreasing, or constant.

Which of the following statements is/are correct? (S1) A certain progression is a geometric progression with a common ratio \(\frac{2}{3}\). (S2) The progression is an increasing sequence with a positive common ratio.

This problem involves analyzing a geometric progression (GP) with a common ratio of 2/3. The solution explains why the sequence decreases when the common ratio is less than 1. Suitable for students in Grades 9-11.

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