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.