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Let's walk through the problem to determine if the given series converges or diverges, and if it converges, find the sum.

### Given:
\[
\sum_{n=1}^{\infty} \frac{3^n + 4^n}{6^n}
\]

### Step 1: Rewrite the series
The term inside the summation can be split:
\[
\frac{3^n + 4^n}{6^n} = \frac{3^n}{6^n} + \frac{4^n}{6^n} = \left(\frac{3}{6}\right)^n + \left(\frac{4}{6}\right)^n
\]

This simplifies to:
\[
\left(\frac{1}{2}\right)^n + \left(\frac{2}{3}\right)^n
\]

### Step 2: Analyze the two series
Both components are geometric series:

- **For** \(\left(\frac{1}{2}\right)^n\):
  - This is a geometric series with \(r = \frac{1}{2}\). Since \(|r| < 1\), it **converges**.
  - Sum of the series:
    \[
    \frac{1/2}{1 - 1/2} = 1
    \]

- **For** \(\left(\frac{2}{3}\right)^n\):
  - This is another geometric series with \(r = \frac{2}{3}\). Since \(|r| < 1\), it also **converges**.
  - Sum of the series:
    \[
    \frac{2/3}{1 - 2/3} = 2
    \]

### Step 3: Find the total sum
Since both geometric series converge, the total sum is the sum of their individual sums:
\[
1 + 2 = 3
\]

### Step 4: Conclusion
The given series **converges** and its sum is \(3\).

---

### Correct Answer:
The correct choice is **A**:
- "The series converges because it is the sum of two geometric series, each with \(|r| < 1\)."  
- The sum of the series is:
  \[
  3
  \]

---

Do you have any questions, or would you like further clarification?

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Here are **five related questions** for further exploration:
1. What are the conditions for a geometric series to converge?
2. How would the sum change if one of the terms had \(|r| \geq 1\)?
3. Can you generalize the sum of a series like \(\sum_{n=1}^{\infty} \frac{a^n + b^n}{c^n}\)?
4. What is the relationship between convergence tests like the Ratio Test and geometric series?
5. How does changing the base of the exponents (e.g., \(5^n\) or \(7^n\)) affect convergence?

---

**Tip:** Always check if you can split complex series into simpler parts, such as sums of geometric series, to analyze convergence easily.

Determine whether the series \(\sum_{n=1}^{\infty} \frac{3^n + 4^n}{6^n}\) converges or diverges. If it converges, find its sum.

This problem involves determining whether the series \(\sum_{n=1}^{\infty} \frac{3^n + 4^n}{6^n}\) converges or diverges. The solution demonstrates how to break down the series into two geometric series and apply the geometric series formula to find the sum. A detailed step-by-step explanation is provided.

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