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The problem in the image involves a geometric sequence \(\{a_1, a_2, a_3, \dots\}\), where specific values are given for \(a_2\) and \(a_4\), and we are asked to find the sum of the infinite series \( \sum_{n=1}^{\infty} a_n \).

### Key Information:
1. **Geometric Sequence**: \(a_1, a_2, a_3, \dots\)
2. Given values:
   \[
   a_2 = \frac{4}{25}, \quad a_4 = \frac{16}{625}, \quad r > 0
   \]
3. Find the sum \(S = \sum_{n=1}^{\infty} a_n \).

### Steps to solve:

1. **General Formula of Geometric Sequence**:
   \[
   a_n = a_1 \cdot r^{n-1}
   \]
   where \(r\) is the common ratio.

2. **Use the given values**:
   - From \(a_2 = \frac{4}{25}\), we know:
     \[
     a_2 = a_1 \cdot r = \frac{4}{25}
     \]
     Hence:
     \[
     a_1 \cdot r = \frac{4}{25}
     \]
   - From \(a_4 = \frac{16}{625}\), we know:
     \[
     a_4 = a_1 \cdot r^3 = \frac{16}{625}
     \]
     
3. **Solve for \(r\)**:
   Now, divide the second equation by the first to eliminate \(a_1\):
   \[
   \frac{a_4}{a_2} = \frac{a_1 \cdot r^3}{a_1 \cdot r} = r^2
   \]
   \[
   \frac{\frac{16}{625}}{\frac{4}{25}} = r^2
   \]
   Simplifying the left-hand side:
   \[
   r^2 = \frac{16}{625} \cdot \frac{25}{4} = \frac{16 \times 25}{625 \times 4} = \frac{400}{2500} = \frac{1}{6.25} = 0.16
   \]
   Thus:
   \[
   r = \sqrt{0.16} = 0.4
   \]

4. **Find \(a_1\)**:
   Now that we know \(r = 0.4\), use \(a_2 = a_1 \cdot r\):
   \[
   \frac{4}{25} = a_1 \cdot 0.4
   \]
   \[
   a_1 = \frac{\frac{4}{25}}{0.4} = \frac{4}{25 \times 0.4} = \frac{4}{10} = 0.4
   \]

5. **Sum of the Infinite Geometric Series**:
   The sum \(S\) of an infinite geometric series is given by:
   \[
   S = \frac{a_1}{1 - r}
   \]
   Substituting \(a_1 = 0.4\) and \(r = 0.4\):
   \[
   S = \frac{0.4}{1 - 0.4} = \frac{0.4}{0.6} = \frac{2}{3}
   \]

### Final Answer:
The sum of the infinite series is:
\[
S = \frac{2}{3}
\]

Would you like more details on any step, or have any questions?

Here are 5 related questions:
1. What are the key properties of a geometric sequence?
2. How do you derive the sum of an infinite geometric series?
3. How does the common ratio \(r\) affect the convergence of an infinite series?
4. Can an infinite geometric series converge if \(r \geq 1\)?
5. What is the impact of different initial terms (\(a_1\)) on the sum of the series?

**Tip**: Always check the value of the common ratio \(r\) in a geometric series. For convergence, \(|r| < 1\) is required.

Given a geometric sequence with a_2 = 4/25, a_4 = 16/625, and r > 0, find the sum of the infinite series ∑ a_n where n = 1 to ∞.

This problem explores a geometric sequence where a_2 = 4/25 and a_4 = 16/625. Learn how to find the common ratio and compute the sum of the infinite series. Ideal for students in Grades 10-12.

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