Consider the finite geometric series.
∑
𝑘
=
1
𝑛
4
(
1
3
)
𝑘
−
1
k=1
∑
n
​
4(
3
1
​
)
k−1

Evaluate the series for each of the given values of
𝑛
n​. You may round your answers to five decimal places.

𝑛
=
6
n=6.​
Preview
​
𝑛
=
8
n=8​.
Preview
​
𝑛
=
10
n=10​.

To evaluate the given finite geometric series for different values of \( n \), we can use the formula for the sum of a geometric series:

\[ S_n = a \frac{1 - r^n}{1 - r} \]

where:
- \( a \) is the first term of the series,
- \( r \) is the common ratio, and
- \( n \) is the number of terms.

For the series \( \sum_{k=1}^{n} 4\left(\frac{1}{3}\right)^{k-1} \), the first term \( a \) is 4 (when \( k = 1 \)) and the common ratio \( r \) is \( \frac{1}{3} \).

Let's calculate the sum for each value of \( n \).

1. **For \( n = 6 \)**:
   - \( a = 4 \)
   - \( r = \frac{1}{3} \)
   - \( n = 6 \)

   \[
   S_6 = 4 \frac{1 - \left(\frac{1}{3}\right)^6}{1 - \frac{1}{3}}
   \]

2. **For \( n = 8 \)**:
   - \( a = 4 \)
   - \( r = \frac{1}{3} \)
   - \( n = 8 \)

   \[
   S_8 = 4 \frac{1 - \left(\frac{1}{3}\right)^8}{1 - \frac{1}{3}}
   \]

3. **For \( n = 10 \)**:
   - \( a = 4 \)
   - \( r = \frac{1}{3} \)
   - \( n = 10 \)

   \[
   S_{10} = 4 \frac{1 - \left(\frac{1}{3}\right)^{10}}{1 - \frac{1}{3}}
   \]

Let's calculate these values.Here are the evaluated sums for the geometric series with different values of \( n \), rounded to five decimal places:

1. For \( n = 6 \): \( S_6 \approx 5.99177 \)
2. For \( n = 8 \): \( S_8 \approx 5.99909 \)
3. For \( n = 10 \): \( S_{10} \approx 5.99990 \)

Would you like any further details or explanations?

Here are some related questions to consider:
1. How does the common ratio affect the convergence of a geometric series?
2. What happens to the sum as \( n \) approaches infinity for this series?
3. How would the sum change if the first term \( a \) were different?
4. What is the significance of the common ratio being less than 1?
5. Can a geometric series have a negative common ratio? If so, how does it affect the series?
6. What if the common ratio was greater than 1? How would that affect the series?
7. How can we use geometric series in real-world applications?
8. What are the conditions for a geometric series to converge?

**Tip:** Understanding the behavior of a geometric series as the number of terms increases can help in approximating the sum for large \( n \).

Learn how to evaluate the finite geometric series \( \sum_{k=1}^{n} 4\left(\frac{1}{3}\right)^{k-1} \) for different values of \( n \). Detailed step-by-step calculation and answers rounded to five decimal places. Suitable for advanced high school students studying geometric series and sequences.

Math Problem Statement

Consider the finite geometric series. ∑ 𝑘

𝑛

6 n=6. Preview 𝑛

8 n=8. Preview 𝑛

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