In the context of a geometric series, \( r \) is called the **common ratio**. It represents the factor by which each term in the series is multiplied to get the next term.

For the series \(\sum_{k=1}^{n} 2^k\), each term is of the form \(2^k\). The common ratio \(r\) is the factor that you multiply one term by to get the next term. 

Given that the terms in the series are \(2^1, 2^2, 2^3, \dots, 2^n\), you can see that:

\[
\frac{2^2}{2^1} = \frac{2^3}{2^2} = \frac{2^4}{2^3} = \dots = 2
\]

Thus, in this series, the common ratio \( r = 2 \).

Would you like further clarification, or do you have any more questions?

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**Related Questions:**

1. How do you find the common ratio in a geometric series?
2. What is the difference between an arithmetic and a geometric series?
3. Can the common ratio \( r \) be a negative number?
4. How does the value of \( r \) affect the sum of a geometric series?
5. What is the common ratio in the series \( \sum_{k=1}^{n} 3^k \)?

**Tip:** The common ratio \( r \) is crucial in determining the behavior of a geometric series, especially when calculating its sum or analyzing its convergence.

Explore the concept of the common ratio (r) in geometric series with detailed explanations and examples. Learn how the common ratio affects the progression of terms in a geometric series, and its significance in determining series behavior and convergence. Ideal for high school students studying sequences and series in mathematics.

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