Finding the Correct Summation Formula

Oct 13, 2025 by miodeya.com 122 views

Introduction

In mathematics, summation formulas are used to represent the sum of a series of numbers. These formulas are essential in various mathematical operations, including algebra, calculus, and statistics. In this article, we will explore the summation formula that represents the given series: 13 + 9 + 5 + 1.

Understanding the Series

The given series is a simple arithmetic sequence with a common difference of -4. The first term is 13, and the last term is 1. To find the summation formula, we need to understand the pattern of the series.

Analyzing the Options

We are given four options to choose from:

A. $\sum_{n=1}^4(-4n + 17)$ B. $\sum_{n=1}^4(-4n - 15)$ C. $\sum_{n=13}^{16}(n - 4)$ D. $\sum_{n=17}^{20}(n - 4)$

To determine the correct option, we need to evaluate each formula and compare the result with the given series.

Evaluating Option A

Let's start by evaluating option A: $\sum_{n=1}^4(-4n + 17)$ .

$\sum_{n=1}^4(-4n + 17) = (-4(1) + 17) + (-4(2) + 17) + (-4(3) + 17) + (-4(4) + 17)$

$= (13) + (9) + (5) + (1)$

$= 28$

This result is not equal to the given series, so option A is incorrect.

Evaluating Option B

Next, let's evaluate option B: $\sum_{n=1}^4(-4n - 15)$ .

$\sum_{n=1}^4(-4n - 15) = (-4(1) - 15) + (-4(2) - 15) + (-4(3) - 15) + (-4(4) - 15)$

$= (-19) + (-31) + (-43) + (-55)$

$= -148$

This result is not equal to the given series, so option B is incorrect.

Evaluating Option C

Now, let's evaluate option C: $\sum_{n=13}^{16}(n - 4)$ .

$\sum_{n=13}^{16}(n - 4) = (13 - 4) + (14 - 4) + (15 - 4) + (16 - 4)$

$= (9) + (10) + (11) + (12)$

$= 42$

This result is not equal to the given series, so option C is incorrect.

Evaluating Option D

Finally, let's evaluate option D: $\sum_{n=17}^{20}(n - 4)$ .

$\sum_{n=17}^{20}(n - 4) = (17 - 4) + (18 - 4) + (19 - 4) + (20 - 4)$

$= (13) + (14) + (15) + (16)$

$= 58$

This result is not equal to the given series, so option D is incorrect.

Conclusion

After evaluating all the options, we can conclude that none of the given options represent the series 13 + 9 + 5 + 1. However, we can try to find the correct summation formula by analyzing the pattern of the series.

Finding the Correct Summation Formula

The given series is an arithmetic sequence with a common difference of -4. The first term is 13, and the last term is 1. We can use the formula for the sum of an arithmetic series:

$\sum_{n=1}^4 a_n = \frac{n}{2}(a_1 + a_n)$

where $a_n$ is the nth term of the series, and $n$ is the number of terms.

In this case, $a_1 = 13$ , $a_n = 1$ , and $n = 4$ . Plugging these values into the formula, we get:

$\sum_{n=1}^4 a_n = \frac{4}{2}(13 + 1)$

$= 2(14)$

$= 28$

However, this result is not equal to the given series. We need to find the correct summation formula.

The Correct Summation Formula

After analyzing the pattern of the series, we can find the correct summation formula:

$\sum_{n=1}^4 (17 - 4n)$

This formula represents the series 13 + 9 + 5 + 1.

Conclusion

In conclusion, the correct summation formula that represents the series 13 + 9 + 5 + 1 is $\sum_{n=1}^4 (17 - 4n)$ . This formula uses the arithmetic sequence with a common difference of -4 and the first term of 13 to represent the series.

Final Answer

$The final answer is $\boxed{\sum_{n=1}^4 (17 - 4n)}$.<br/> **Solving the Summation Formula: A Step-by-Step Guide**$

Q&A: Summation Formulas and Arithmetic Sequences

Introduction

In our previous article, we explored the summation formula that represents the series 13 + 9 + 5 + 1. We found that the correct summation formula is $\sum_{n=1}^4 (17 - 4n)$ . In this article, we will answer some frequently asked questions about summation formulas and arithmetic sequences.

Q: What is a summation formula?

A: A summation formula is a mathematical expression that represents the sum of a series of numbers. It is used to calculate the total value of a sequence of numbers.

Q: What is an arithmetic sequence?

A: An arithmetic sequence is a sequence of numbers in which each term is obtained by adding a fixed constant to the previous term. The common difference between consecutive terms is constant.

Q: How do I find the summation formula for an arithmetic sequence?

A: To find the summation formula for an arithmetic sequence, you need to know the first term, the last term, and the number of terms. You can use the formula for the sum of an arithmetic series:

$\sum_{n=1}^n a_n = \frac{n}{2}(a_1 + a_n)$

where $a_n$ is the nth term of the series, and $n$ is the number of terms.

Q: What is the formula for the nth term of an arithmetic sequence?

A: The formula for the nth term of an arithmetic sequence is:

$a_n = a_1 + (n - 1)d$

where $a_n$ is the nth term, $a_1$ is the first term, $n$ is the term number, and $d$ is the common difference.

Q: How do I find the common difference of an arithmetic sequence?

A: To find the common difference of an arithmetic sequence, you need to know the first term and the second term. You can use the formula:

$d = a_2 - a_1$

where $d$ is the common difference, $a_2$ is the second term, and $a_1$ is the first term.

Q: What is the formula for the sum of an arithmetic series?

A: The formula for the sum of an arithmetic series is:

$\sum_{n=1}^n a_n = \frac{n}{2}(a_1 + a_n)$

where $a_n$ is the nth term of the series, and $n$ is the number of terms.

Q: How do I use the formula for the sum of an arithmetic series?

A: To use the formula for the sum of an arithmetic series, you need to know the first term, the last term, and the number of terms. You can plug these values into the formula to find the sum of the series.

Q: What are some common applications of summation formulas?

A: Summation formulas have many applications in mathematics, science, and engineering. Some common applications include:

Calculating the total value of a sequence of numbers
Finding the sum of an arithmetic series
Solving problems involving finance, economics, and statistics
Modeling real-world phenomena, such as population growth and chemical reactions

Conclusion

In conclusion, summation formulas and arithmetic sequences are fundamental concepts in mathematics. By understanding these concepts, you can solve a wide range of problems and apply mathematical techniques to real-world situations.

Final Answer

The final answer is $\boxed{\sum_{n=1}^4 (17 - 4n)}$ .