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To find the sum of the series \(\sum_{k=1}^\infty \frac{3n - 2}{5^n}\), we will approach it by splitting it into two separate sums and then finding each of those sums individually. 

Let's rewrite the given series as:
\[
\sum_{n=1}^\infty \frac{3n - 2}{5^n} = \sum_{n=1}^\infty \frac{3n}{5^n} - \sum_{n=1}^\infty \frac{2}{5^n}
\]

First, we evaluate the second sum:
\[
\sum_{n=1}^\infty \frac{2}{5^n}
\]
This is a geometric series with the first term \(a = \frac{2}{5}\) and common ratio \(r = \frac{1}{5}\). The sum of an infinite geometric series is given by:
\[
S = \frac{a}{1 - r} = \frac{\frac{2}{5}}{1 - \frac{1}{5}} = \frac{\frac{2}{5}}{\frac{4}{5}} = \frac{2}{4} = \frac{1}{2}
\]

Now, we need to evaluate the first sum:
\[
\sum_{n=1}^\infty \frac{3n}{5^n}
\]
We use the formula for the sum of a series involving \(n\) multiplied by a geometric progression:
\[
\sum_{n=1}^\infty \frac{n}{5^n} = \frac{r}{(1 - r)^2}
\]
where \(r = \frac{1}{5}\). Therefore:
\[
\sum_{n=1}^\infty \frac{n}{5^n} = \frac{\frac{1}{5}}{(1 - \frac{1}{5})^2} = \frac{\frac{1}{5}}{(\frac{4}{5})^2} = \frac{\frac{1}{5}}{\frac{16}{25}} = \frac{25}{20} = \frac{5}{4}
\]

Therefore, we have:
\[
\sum_{n=1}^\infty \frac{3n}{5^n} = 3 \times \frac{5}{4} = \frac{15}{4}
\]

Putting it all together:
\[
\sum_{n=1}^\infty \frac{3n - 2}{5^n} = \frac{15}{4} - \frac{1}{2} = \frac{15}{4} - \frac{2}{4} = \frac{13}{4}
\]

Thus, the sum of the series is:
\[
\boxed{\frac{13}{4}}
\]

Do you want more details or have any questions?

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:** When dealing with series, breaking them down into simpler parts can often simplify the problem-solving process.

Learn how to find the sum of the series \(\sum_{n=1}^\infty \frac{3n - 2}{5^n}\) step-by-step. This problem involves breaking down the series into simpler parts, calculating geometric series sums, and applying formulas for series involving geometric progressions. Suitable for advanced high school students.

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