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This sequence appears to be an infinite geometric series:

\[
\frac{2}{5} + \frac{2}{25} + \frac{2}{125} + \dots
\]

To solve for the sum of this series, we need to identify the first term \( a \) and the common ratio \( r \).

### Step 1: Identify \( a \) and \( r \)
1. **First term \( a \):** The first term is \(\frac{2}{5}\).
2. **Common ratio \( r \):** To find \( r \), divide the second term by the first term:
   \[
   r = \frac{\frac{2}{25}}{\frac{2}{5}} = \frac{2}{25} \times \frac{5}{2} = \frac{1}{5}
   \]

### Step 2: Use the Sum Formula for an Infinite Geometric Series
The sum \( S \) of an infinite geometric series with \(|r| < 1\) is given by:
\[
S = \frac{a}{1 - r}
\]

### Step 3: Substitute \( a \) and \( r \) into the Formula
\[
S = \frac{\frac{2}{5}}{1 - \frac{1}{5}} = \frac{\frac{2}{5}}{\frac{4}{5}} = \frac{2}{5} \times \frac{5}{4} = \frac{1}{2}
\]

### Final Answer
The sum of the series is:
\[
\boxed{\frac{1}{2}}
\]

Would you like a deeper explanation or have any questions?

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**Further Questions for Exploration:**
1. How do you determine whether an infinite series converges or diverges?
2. Can an infinite geometric series with \( |r| \geq 1 \) ever have a finite sum?
3. What happens if the first term \( a \) is negative?
4. How does changing \( r \) affect the rate of convergence of the series?
5. How is the sum formula for an infinite geometric series derived?

**Tip:** When dealing with an infinite geometric series, always check that \( |r| < 1 \) to ensure convergence.

Explore how to solve an infinite geometric series with the initial terms 2/5, 2/25, and 2/125. This solution includes identifying the first term and common ratio, applying the sum formula for infinite geometric series, and determining convergence. Suitable for high school students studying series.

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