The Formula Behind the Series

Oct 13, 2025 by talynt.org 135 views

Introduction

When it comes to evaluating series, mathematicians often rely on various formulas and techniques to simplify the process. In this article, we will explore the evaluation of the series $\sum_{k=1}^9 2(4)^{k-1}$ and examine the options provided to determine which one can be used to evaluate the 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 series is $\sum_{k=1}^9 2(4)^{k-1}$ , where the first term is $2(4)^{0}$ and the common ratio is $4$ . The series has a total of $9$ terms.

Evaluating the Series

To evaluate the series, we can use the formula for the sum of a geometric series:

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.

Option A: $2\left(\frac{1-4^8}{1-4}\right)$

Let's examine option A: $2\left(\frac{1-4^8}{1-4}\right)$ . This expression can be simplified as follows:

2\left(\frac{1-4^8}{1-4}\right) = 2\left(\frac{1-65536}{-3}\right) = 2\left(\frac{-65535}{-3}\right) = 2(21845) = 43690

Option B: $2\left(\frac{1-4^{10}}{1-4}\right)$

Now, let's examine option B: $2\left(\frac{1-4^{10}}{1-4}\right)$ . This expression can be simplified as follows:

2\left(\frac{1-4^{10}}{1-4}\right) = 2\left(\frac{1-1048576}{-3}\right) = 2\left(\frac{-1048575}{-3}\right) = 2(349525) = 699050

Option C: $2\left(\frac{1-4^9}{1-4}\right)$

Finally, let's examine option C: $2\left(\frac{1-4^9}{1-4}\right)$ . This expression can be simplified as follows:

2\left(\frac{1-4^9}{1-4}\right) = 2\left(\frac{1-262144}{-3}\right) = 2\left(\frac{-262143}{-3}\right) = 2(87381) = 174762

Conclusion

Based on the calculations above, we can see that option C: $2\left(\frac{1-4^9}{1-4}\right)$ is the correct answer. This expression can be used to evaluate the series $\sum_{k=1}^9 2(4)^{k-1}$ .

The Formula Behind the Series

The formula behind the series is a geometric series, which is a type of series that has a constant ratio between consecutive terms. In this case, the series is $\sum_{k=1}^9 2(4)^{k-1}$ , where the first term is $2(4)^{0}$ and the common ratio is $4$ . The series has a total of $9$ terms.

The Formula for the Sum of a Geometric Series

The formula for the sum of a geometric series is:

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.

Using the Formula to Evaluate the Series

To evaluate the series, we can use the formula for the sum of a geometric series. Plugging in the values, we get:

S_9 = \frac{2(1-4^9)}{1-4} = \frac{2(1-262144)}{-3} = \frac{2(-262143)}{-3} = 2(87381) = 174762

Conclusion

Based on the calculations above, we can see that the correct answer is option C: $2\left(\frac{1-4^9}{1-4}\right)$ . This expression can be used to evaluate the series $\sum_{k=1}^9 2(4)^{k-1}$ .

The Importance of Understanding the Formula

Understanding the formula for the sum of a geometric series is crucial in evaluating series. The formula provides a simple and efficient way to calculate the sum of a series, especially when the series has a large number of terms.

The Formula as a Tool for Problem-Solving

The formula for the sum of a geometric series can be used as a tool for problem-solving. By applying the formula, we can evaluate series and solve problems that involve geometric series.

Conclusion

In conclusion, the correct answer is option C: $2\left(\frac{1-4^9}{1-4}\right)$ . This expression can be used to evaluate the series $\sum_{k=1}^9 2(4)^{k-1}$ . Understanding the formula for the sum of a geometric series is crucial in evaluating series, and the formula can be used as a tool for problem-solving.

Introduction

Understanding the Series

Evaluating the Series

To evaluate the series, we can use the formula for the sum of a geometric series:

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.

Option A: $2\left(\frac{1-4^8}{1-4}\right)$

Let's examine option A: $2\left(\frac{1-4^8}{1-4}\right)$ . This expression can be simplified as follows:

2\left(\frac{1-4^8}{1-4}\right) = 2\left(\frac{1-65536}{-3}\right) = 2\left(\frac{-65535}{-3}\right) = 2(21845) = 43690

Option B: $2\left(\frac{1-4^{10}}{1-4}\right)$

Now, let's examine option B: $2\left(\frac{1-4^{10}}{1-4}\right)$ . This expression can be simplified as follows:

2\left(\frac{1-4^{10}}{1-4}\right) = 2\left(\frac{1-1048576}{-3}\right) = 2\left(\frac{-1048575}{-3}\right) = 2(349525) = 699050

Option C: $2\left(\frac{1-4^9}{1-4}\right)$

Finally, let's examine option C: $2\left(\frac{1-4^9}{1-4}\right)$ . This expression can be simplified as follows:

2\left(\frac{1-4^9}{1-4}\right) = 2\left(\frac{1-262144}{-3}\right) = 2\left(\frac{-262143}{-3}\right) = 2(87381) = 174762

Conclusion

Based on the calculations above, we can see that option C: $2\left(\frac{1-4^9}{1-4}\right)$ is the correct answer. This expression can be used to evaluate the series $\sum_{k=1}^9 2(4)^{k-1}$ .

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.

Q: What is the formula for the sum of a geometric series?

A: The formula for the sum of a geometric series is:

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.

Q: How do I evaluate a geometric series?

A: To evaluate a geometric series, you can use the formula for the sum of a geometric series. Plug in the values of the first term, common ratio, and number of terms, and simplify the expression.

Q: What is the correct answer for the series $\sum_{k=1}^9 2(4)^{k-1}$ ?

A: The correct answer is option C: $2\left(\frac{1-4^9}{1-4}\right)$ .

Q: Why is understanding the formula for the sum of a geometric series important?

A: Understanding the formula for the sum of a geometric series is important because it provides a simple and efficient way to calculate the sum of a series, especially when the series has a large number of terms.

Q: Can I use the formula for the sum of a geometric series to evaluate any series?

A: No, the formula for the sum of a geometric series can only be used to evaluate geometric series. If the series is not geometric, you will need to use a different formula or technique to evaluate it.

Q: What is the significance of the common ratio in a geometric series?

A: The common ratio in a geometric series is the ratio between consecutive terms. It determines the rate at which the terms of the series increase or decrease.

Q: How do I determine the number of terms in a geometric series?

A: The number of terms in a geometric series is typically given in the problem statement. If it is not given, you will need to determine it based on the context of the problem.

Q: Can I use the formula for the sum of a geometric series to evaluate a series with a negative common ratio?

A: Yes, the formula for the sum of a geometric series can be used to evaluate a series with a negative common ratio. However, you will need to take into account the sign of the common ratio when simplifying the expression.

Q: What is the relationship between the sum of a geometric series and the formula for the sum of a geometric series?

A: The sum of a geometric series is equal to the formula for the sum of a geometric series, which is:

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.