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The problem shows a geometric sequence \(a_1, a_2, a_3, \ldots, a_n\) where:

- \( a_2 = \frac{4}{25} \)
- \( a_4 = \frac{16}{625} \)
- \( r > 0 \) is the common ratio
- You are asked to find the infinite sum \( \sum_{n=1}^{\infty} a_n \)

### Step 1: Find the common ratio \(r\)

Since this is a geometric sequence, the relationship between terms is given by:

\[
a_n = a_1 \cdot r^{n-1}
\]

Using the known values of \( a_2 \) and \( a_4 \):

\[
a_2 = a_1 \cdot r = \frac{4}{25}
\]

\[
a_4 = a_1 \cdot r^3 = \frac{16}{625}
\]

We now have the system of equations:

\[
a_1 \cdot r = \frac{4}{25}
\]
\[
a_1 \cdot r^3 = \frac{16}{625}
\]

Dividing the second equation by the first to eliminate \(a_1\):

\[
\frac{a_1 \cdot r^3}{a_1 \cdot r} = \frac{\frac{16}{625}}{\frac{4}{25}}
\]

Simplifying:

\[
r^2 = \frac{16}{625} \times \frac{25}{4} = \frac{16 \times 25}{625 \times 4} = \frac{400}{2500} = \frac{4}{25}
\]

Thus:

\[
r = \frac{2}{5}
\]

### Step 2: Find \(a_1\)

Now substitute \(r = \frac{2}{5}\) into the equation for \(a_2\):

\[
a_1 \cdot \frac{2}{5} = \frac{4}{25}
\]

Solving for \(a_1\):

\[
a_1 = \frac{4}{25} \cdot \frac{5}{2} = \frac{20}{50} = \frac{2}{5}
\]

### Step 3: Find the infinite sum

The sum of an infinite geometric series is given by:

\[
S = \frac{a_1}{1 - r}
\]

Substitute \(a_1 = \frac{2}{5}\) and \(r = \frac{2}{5}\):

\[
S = \frac{\frac{2}{5}}{1 - \frac{2}{5}} = \frac{\frac{2}{5}}{\frac{3}{5}} = \frac{2}{5} \cdot \frac{5}{3} = \frac{2}{3}
\]

### Final Answer:
The infinite sum of the geometric sequence is:

\[
\boxed{\frac{2}{3}}
\]

---

Would you like further explanation on any step? Here are some related questions:

1. How do you derive the sum formula for an infinite geometric series?
2. Can the sum formula be applied if \(r \geq 1\)?
3. What happens to the series if \(r < 0\)?
4. How would the sum change if the first term were different?
5. Could you solve for any general term \(a_n\) in this sequence?

**Tip:** Always verify that the common ratio \(r\) satisfies \( |r| < 1 \) when calculating the infinite sum, otherwise, the series diverges.

Find the infinite sum of the geometric sequence where a_2 = 4/25 and a_4 = 16/625 with r > 0.

Learn how to find the infinite sum of a geometric sequence where the second term is 4/25 and the fourth term is 16/625. This problem involves key concepts of geometric sequences and series, including finding the common ratio and the first term. Suitable for students in Grades 9-12.

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