Let's go through the solution step-by-step again to identify the error and correct it.

The series given is:
\[
\sum_{k=1}^\infty \frac{3n - 2}{5^n}
\]

First, let's correct the indices and properly handle each part of the series separately.

### Step 1: Split the Series
\[
\sum_{n=1}^\infty \frac{3n - 2}{5^n} = \sum_{n=1}^\infty \frac{3n}{5^n} - \sum_{n=1}^\infty \frac{2}{5^n}
\]

### Step 2: Evaluate Each Series Separately

#### Sum 1: \(\sum_{n=1}^\infty \frac{2}{5^n}\)

This is a geometric series with the first term \(a = \frac{2}{5}\) and common ratio \(r = \frac{1}{5}\):
\[
\sum_{n=1}^\infty \frac{2}{5^n} = \frac{\frac{2}{5}}{1 - \frac{1}{5}} = \frac{\frac{2}{5}}{\frac{4}{5}} = \frac{2}{4} = \frac{1}{2}
\]

#### Sum 2: \(\sum_{n=1}^\infty \frac{3n}{5^n}\)

We use the formula for the sum of a series involving \(n\) multiplied by a geometric progression:
\[
\sum_{n=1}^\infty \frac{n}{5^n} = \frac{r}{(1 - r)^2} \quad \text{where} \quad r = \frac{1}{5}
\]

Thus:
\[
\sum_{n=1}^\infty \frac{n}{5^n} = \frac{\frac{1}{5}}{(1 - \frac{1}{5})^2} = \frac{\frac{1}{5}}{\left(\frac{4}{5}\right)^2} = \frac{\frac{1}{5}}{\frac{16}{25}} = \frac{25}{80} = \frac{5}{16}
\]

Now, multiply this result by 3:
\[
\sum_{n=1}^\infty \frac{3n}{5^n} = 3 \times \frac{5}{16} = \frac{15}{16}
\]

### Step 3: Combine the Results
\[
\sum_{n=1}^\infty \frac{3n - 2}{5^n} = \frac{15}{16} - \frac{1}{2} = \frac{15}{16} - \frac{8}{16} = \frac{7}{16}
\]

Thus, the correct sum of the series is:
\[
\boxed{\frac{7}{16}}
\]

Do you have any questions or would you like more details on any step?

Here are 5 more questions you can ask:
1. How do you find the sum of a geometric series?
2. Can you explain the steps to derive the sum formula for \(\sum_{n=1}^\infty \frac{n}{5^n}\)?
3. What are other methods to solve series involving factorials?
4. How do you approach solving an alternating series?
5. Can you explain the concept of convergence in series?

**Tip:** Carefully checking each step of a series evaluation can help catch and correct errors in the calculation.

Learn how to correctly evaluate the series \(\sum_{n=1}^\infty \frac{3n - 2}{5^n}\) step-by-step with a detailed explanation. This problem covers key concepts such as geometric series, infinite series, and summation techniques. Suitable for advanced high school students and anyone interested in series evaluation.

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