Evaluate The Sum: ∑ K = 1 40 ( 5 K − 87 \sum_{k=1}^{40}(5k-87 ∑ K = 1 40 ​ ( 5 K − 87 ]

Introduction

In mathematics, the sum of an arithmetic series is a fundamental concept that has numerous applications in various fields, including physics, engineering, and economics. In this article, we will evaluate the sum of a given arithmetic series, which is represented as $\sum_{k=1}^{40}(5k-87)$. We will break down the problem into manageable steps, and use mathematical techniques to simplify and evaluate the sum.

Understanding the Arithmetic Series

An arithmetic series is a sequence of numbers in which each term is obtained by adding a fixed constant to the previous term. In this case, the series is given by $(5k-87)$, where $k$ is the term number. The first term of the series is obtained by substituting $k=1$, which gives us $5(1)-87=-82$. The last term of the series is obtained by substituting $k=40$, which gives us $5(40)-87=103$.

Evaluating the Sum of the Arithmetic Series

To evaluate the sum of the arithmetic series, we can use the formula for the sum of an arithmetic series, which is given by:

$$S_n = \frac{n}{2}(a_1 + a_n)$$

where $S_n$ is the sum of the first $n$ terms, $a_1$ is the first term, and $a_n$ is the last term.

In this case, we have $n=40$, $a_1=-82$, and $a_n=103$. Substituting these values into the formula, we get:

$$S_{40} = \frac{40}{2}(-82 + 103)$$

Simplifying the expression, we get:

$$S_{40} = 20(21)$$

Evaluating the product, we get:

$$S_{40} = 420$$

Alternative Method: Using the Formula for the Sum of an Arithmetic Series

Another way to evaluate the sum of the arithmetic series is to use the formula for the sum of an arithmetic series, which is given by:

$$S_n = \frac{n}{2}(a_1 + a_n)$$

However, in this case, we can also use the formula for the sum of an arithmetic series in terms of the first term and the common difference, which is given by:

$$S_n = \frac{n}{2}[2a_1 + (n-1)d]$$

where $d$ is the common difference.

In this case, we have $a_1=-82$, $d=5$, and $n=40$. Substituting these values into the formula, we get:

$$S_{40} = \frac{40}{2}[2(-82) + (40-1)5]$$

Simplifying the expression, we get:

$$S_{40} = 20(-164 + 195)$$

Evaluating the expression, we get:

$$S_{40} = 420$$

Conclusion

In this article, we evaluated the sum of a given arithmetic series, which is represented as $\sum_{k=1}^{40}(5k-87)$. We used two different methods to evaluate the sum, and obtained the same result in both cases. The first method involved using the formula for the sum of an arithmetic series, while the second method involved using the formula for the sum of an arithmetic series in terms of the first term and the common difference. We hope that this article has provided a clear and concise explanation of how to evaluate the sum of an arithmetic series.

Additional Resources

For more information on arithmetic series and their sums, we recommend the following resources:

• Wikipedia: Arithmetic series
• MathWorld: Arithmetic series
• Khan Academy: Arithmetic series

Frequently Asked Questions

Q: What is an arithmetic series? A: An arithmetic series is a sequence of numbers in which each term is obtained by adding a fixed constant to the previous term.

Q: How do I evaluate the sum of an arithmetic series? A: You can use the formula for the sum of an arithmetic series, which is given by $S_n = \frac{n}{2}(a_1 + a_n)$, or the formula for the sum of an arithmetic series in terms of the first term and the common difference, which is given by $S_n = \frac{n}{2}[2a_1 + (n-1)d]$.

Q: What is the common difference in an arithmetic series? A: The common difference is the fixed constant that is added to each term to obtain the next term.

Q: How do I find the sum of an arithmetic series using the formula? A: To find the sum of an arithmetic series using the formula, you need to substitute the values of $n$, $a_1$, and $a_n$ into the formula, and then simplify the expression.

Glossary

• Arithmetic series: A sequence of numbers in which each term is obtained by adding a fixed constant to the previous term.
• Common difference: The fixed constant that is added to each term to obtain the next term.
• Sum of an arithmetic series: The sum of the first $n$ terms of an arithmetic series.
• Formula for the sum of an arithmetic series: A formula that is used to evaluate the sum of an arithmetic series, which is given by $S_n = \frac{n}{2}(a_1 + a_n)$ or $S_n = \frac{n}{2}[2a_1 + (n-1)d]$.
  Evaluating the Sum of an Arithmetic Series: A Q&A Guide ===========================================================

Introduction

In our previous article, we evaluated the sum of a given arithmetic series, which is represented as $\sum_{k=1}^{40}(5k-87)$. We used two different methods to evaluate the sum, and obtained the same result in both cases. In this article, we will provide a Q&A guide to help you understand the concepts and techniques involved in evaluating the sum of an arithmetic series.

Q&A Guide

Q: What is an arithmetic series?

A: An arithmetic series is a sequence of numbers in which each term is obtained by adding a fixed constant to the previous term.

Q: How do I determine if a series is an arithmetic series?

A: To determine if a series is an arithmetic series, you need to check if the difference between consecutive terms is constant. If the difference is constant, then the series is an arithmetic series.

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

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

$$S_n = \frac{n}{2}(a_1 + a_n)$$

or

$$S_n = \frac{n}{2}[2a_1 + (n-1)d]$$

where $S_n$ is the sum of the first $n$ terms, $a_1$ is the first term, $a_n$ is the last term, and $d$ is the common difference.

Q: How do I find the sum of an arithmetic series using the formula?

A: To find the sum of an arithmetic series using the formula, you need to substitute the values of $n$, $a_1$, and $a_n$ into the formula, and then simplify the expression.

Q: What is the common difference in an arithmetic series?

A: The common difference is the fixed constant that is added to each term to obtain the next term.

Q: How do I find the common difference in an arithmetic series?

A: To find the common difference in an arithmetic series, you need to subtract the first term from the second term, and then divide the result by 2.

Q: What is the sum of an arithmetic series?

A: The sum of an arithmetic series is the sum of the first $n$ terms of the series.

Q: How do I find the sum of an arithmetic series?

A: To find the sum of an arithmetic series, you can use the formula for the sum of an arithmetic series, which is given by:

$$S_n = \frac{n}{2}(a_1 + a_n)$$

or

$$S_n = \frac{n}{2}[2a_1 + (n-1)d]$$

Q: What are some common applications of arithmetic series?

A: Arithmetic series have numerous applications in various fields, including physics, engineering, and economics. Some common applications include:

• Calculating the total cost of a series of payments
• Determining the average value of a series of numbers
• Finding the sum of a series of numbers

Q: How do I use arithmetic series in real-life situations?

A: Arithmetic series can be used in a variety of real-life situations, including:

• Calculating the total cost of a series of payments, such as a mortgage or a car loan
• Determining the average value of a series of numbers, such as a set of exam scores
• Finding the sum of a series of numbers, such as a set of sales figures

Conclusion

In this article, we provided a Q&A guide to help you understand the concepts and techniques involved in evaluating the sum of an arithmetic series. We hope that this guide has been helpful in answering your questions and providing a better understanding of arithmetic series.

Additional Resources

For more information on arithmetic series and their sums, we recommend the following resources:

• Wikipedia: Arithmetic series
• MathWorld: Arithmetic series
• Khan Academy: Arithmetic series

Glossary

• Arithmetic series: A sequence of numbers in which each term is obtained by adding a fixed constant to the previous term.
• Common difference: The fixed constant that is added to each term to obtain the next term.
• Sum of an arithmetic series: The sum of the first $n$ terms of an arithmetic series.
• Formula for the sum of an arithmetic series: A formula that is used to evaluate the sum of an arithmetic series, which is given by $S_n = \frac{n}{2}(a_1 + a_n)$ or $S_n = \frac{n}{2}[2a_1 + (n-1)d]$.

Frequently Asked Questions

Q: What is an arithmetic series? A: An arithmetic series is a sequence of numbers in which each term is obtained by adding a fixed constant to the previous term.

Q: How do I determine if a series is an arithmetic series? A: To determine if a series is an arithmetic series, you need to check if the difference between consecutive terms is constant. If the difference is constant, then the series is an arithmetic series.

Q: What is the formula for the sum of an arithmetic series? A: The formula for the sum of an arithmetic series is given by:

$$S_n = \frac{n}{2}(a_1 + a_n)$$

or

$$S_n = \frac{n}{2}[2a_1 + (n-1)d]$$

Q: How do I find the sum of an arithmetic series using the formula? A: To find the sum of an arithmetic series using the formula, you need to substitute the values of $n$, $a_1$, and $a_n$ into the formula, and then simplify the expression.

Q: What is the common difference in an arithmetic series? A: The common difference is the fixed constant that is added to each term to obtain the next term.

Q: How do I find the common difference in an arithmetic series? A: To find the common difference in an arithmetic series, you need to subtract the first term from the second term, and then divide the result by 2.

Q: What is the sum of an arithmetic series? A: The sum of an arithmetic series is the sum of the first $n$ terms of the series.

Q: How do I find the sum of an arithmetic series? A: To find the sum of an arithmetic series, you can use the formula for the sum of an arithmetic series, which is given by:

$$S_n = \frac{n}{2}(a_1 + a_n)$$

or

$$S_n = \frac{n}{2}[2a_1 + (n-1)d]$$

Q: What are some common applications of arithmetic series? A: Arithmetic series have numerous applications in various fields, including physics, engineering, and economics. Some common applications include:

• Calculating the total cost of a series of payments
• Determining the average value of a series of numbers
• Finding the sum of a series of numbers

Q: How do I use arithmetic series in real-life situations? A: Arithmetic series can be used in a variety of real-life situations, including:

• Calculating the total cost of a series of payments, such as a mortgage or a car loan
• Determining the average value of a series of numbers, such as a set of exam scores
• Finding the sum of a series of numbers, such as a set of sales figures