Which Of The Following Can Be Used To Evaluate The Series ∑ K = 1 8 5 ( 2 3 ) K − 1 \sum_{k=1}^8 5\left(\frac{2}{3}\right)^{k-1} ∑ K = 1 8 5 ( 3 2 ) K − 1 ?A. 5\left(\frac{1-\left(\frac{2}{3}\right)^8}{1-\frac{2}{3}}\right ]B.

Mar 13, 2025 by ADMIN 231 views

**Evaluating Series: A Comprehensive Approach**

Introduction

When it comes to evaluating series, there are various methods and formulas that can be employed to determine their sum. In this article, we will focus on evaluating the series $\sum_{k=1}^8 5\left(\frac{2}{3}\right)^{k-1}$ using a specific formula. We will explore the different options available and determine which one is the most suitable for this particular series.

Understanding the Series

The given series is a geometric series, which is a type of series that has a constant ratio between consecutive terms. In this case, the ratio is $\frac{2}{3}$ , and the first term is $5$ . The series is defined as:

\sum_{k=1}^8 5\left(\frac{2}{3}\right)^{k-1}

This series can be expanded as:

5 + 5\left(\frac{2}{3}\right) + 5\left(\frac{2}{3}\right)^2 + 5\left(\frac{2}{3}\right)^3 + 5\left(\frac{2}{3}\right)^4 + 5\left(\frac{2}{3}\right)^5 + 5\left(\frac{2}{3}\right)^6 + 5\left(\frac{2}{3}\right)^7

Option A: Using the Formula for Geometric Series

One of the most common methods for evaluating a geometric series is to use the formula:

S_n = \frac{a(1-r^n)}{1-r}

where $S_n$ is the sum of the first $n$ terms, $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms.

In this case, we have:

a = 5

r = \frac{2}{3}

n = 8

Plugging these values into the formula, we get:

S_8 = \frac{5\left(1-\left(\frac{2}{3}\right)^8\right)}{1-\frac{2}{3}}

This is option A.

Option B: Using the Formula for Geometric Series with a Different Approach

Another way to evaluate the series is to use the formula:

S_n = \frac{a(r^n-1)}{r-1}

This formula is similar to the previous one, but it is used when the series starts from the second term.

In this case, we have:

a = 5\left(\frac{2}{3}\right)

r = \frac{2}{3}

n = 7

Plugging these values into the formula, we get:

S_8 = \frac{5\left(\frac{2}{3}\right)\left(\left(\frac{2}{3}\right)^7-1\right)}{\frac{2}{3}-1}

This is option B.

Conclusion

In conclusion, we have evaluated the series $\sum_{k=1}^8 5\left(\frac{2}{3}\right)^{k-1}$ using two different formulas. Option A uses the formula for geometric series, while option B uses a different approach. Both options are valid, but option A is the most straightforward and easiest to use.

Recommendation

Based on our analysis, we recommend using option A to evaluate the series. This is because it is the most straightforward and easiest to use, and it produces the same result as option B.

Final Answer

The final answer is:

\boxed{5\left(\frac{1-\left(\frac{2}{3}\right)^8}{1-\frac{2}{3}}\right)}$<br/> # **Evaluating Series: A Comprehensive Approach - Q&A**

Introduction

In our previous article, we explored the different methods for evaluating the series $\sum_{k=1}^8 5\left(\frac{2}{3}\right)^{k-1}$ . We discussed two options, A and B, and determined that option A is the most straightforward and easiest to use. In this article, we will provide a Q&A section to further clarify any doubts and provide additional information.

Q&A

Q: What is a geometric series?

A: A geometric series is a type of series that has a constant ratio between consecutive terms. In this case, the ratio is $\frac{2}{3}$ , and the first term is $5$ .

Q: What is the formula for evaluating a geometric series?

A: The formula for evaluating a geometric series is: