Consider the finite geometric series.

∑𝑘=362(3)𝑘−1*k*=3∑6​2(3)*k*−1​List each of the 44​ terms from the expanded form of the geometric series. Separate each term with a comma.

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Evaluate the series.

To list each term of the finite geometric series and then evaluate it, we first identify the terms using the series formula:

\[
\sum_{k=3}^{6} 2 \cdot 3^{k-1}
\]

The general term of the series is given by \( a \cdot r^{k-1} \), where \( a = 2 \) (the first term when \( k = 3 \)), \( r = 3 \) (the common ratio), and \( k \) starts from 3 and goes to 6.

### Listing the Terms
To find each term, substitute the values of \( k \) from 3 to 6:

- For \( k = 3 \): \( 2 \cdot 3^{3-1} = 2 \cdot 3^2 = 2 \cdot 9 = 18 \)
- For \( k = 4 \): \( 2 \cdot 3^{4-1} = 2 \cdot 3^3 = 2 \cdot 27 = 54 \)
- For \( k = 5 \): \( 2 \cdot 3^{5-1} = 2 \cdot 3^4 = 2 \cdot 81 = 162 \)
- For \( k = 6 \): \( 2 \cdot 3^{6-1} = 2 \cdot 3^5 = 2 \cdot 243 = 486 \)

The terms are:
18, 54, 162, 486

### Evaluating the Series
To evaluate the series, we sum the terms:

\[ S = 18 + 54 + 162 + 486 \]

Let's calculate the sum.The terms of the geometric series are:

18, 54, 162, 486

The sum of these terms is:

\[ S = 720 \]

Would you like any further details or explanations?

Here are some related questions to consider:
1. What is the formula for the \( n \)-th term of a geometric series?
2. How does the common ratio affect the growth rate of the terms in a geometric series?
3. What happens to the series sum if the number of terms is increased indefinitely?
4. Can a geometric series have negative terms? If so, how does this affect the sum?
5. What is the difference between a finite and infinite geometric series?
6. How do you determine if a geometric series converges or diverges?
7. What are some real-world applications of geometric series?
8. How does changing the first term \( a \) affect the series?

**Tip:** In geometric series, understanding the common ratio's role can provide insights into the series' behavior and sum calculation.

Explore the expansion and evaluation of a finite geometric series 2 * 3^(k-1) where k ranges from 3 to 6. Learn how to list each term, calculate the series sum, and understand the concepts of geometric series. Suitable for high school students.

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