Example of an Arithmetic Series

Oct 13, 2025 by talynt.org 64 views

Arithmetic series are a fundamental concept in mathematics, and they play a crucial role in various mathematical operations. An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant. In this article, we will explore the concept of arithmetic series and learn how to find the sum of its terms.

What is an Arithmetic Series?

An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant. The general form of an arithmetic series is:

a, a + d, a + 2d, a + 3d, ...

where 'a' is the first term, and 'd' is the common difference between the terms.

Example of an Arithmetic Series

Let's consider an example of an arithmetic series: -5, -3, -1, 1, ...

In this series, the first term 'a' is -5, and the common difference 'd' is 2. We can see that each term is obtained by adding 2 to the previous term.

Finding the Sum of the First 21 Terms of the Arithmetic Series

Now, let's find the sum of the first 21 terms of the arithmetic series: -5, -3, -1, 1, ...

To find the sum of the first 21 terms, we can use the formula for the sum of an arithmetic series:

S_n = n/2 * (a + l)

where S_n is the sum of the first n terms, n is the number of terms, a is the first term, and l is the last term.

Step 1: Find the Last Term

To find the last term, we can use the formula for the nth term of an arithmetic series:

a_n = a + (n-1)d

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

In this case, the first term 'a' is -5, the common difference 'd' is 2, and the term number 'n' is 21. Plugging these values into the formula, we get:

a_21 = -5 + (21-1)2 = -5 + 20 * 2 = -5 + 40 = 35

So, the last term is 35.

Step 2: Find the Sum

Now that we have the first term 'a' as -5, the last term 'l' as 35, and the number of terms 'n' as 21, we can plug these values into the formula for the sum of an arithmetic series:

S_21 = 21/2 * (-5 + 35) = 10.5 * 30 = 315

Therefore, the sum of the first 21 terms of the arithmetic series is 315.

Conclusion

In this article, we learned about arithmetic series and how to find the sum of its terms. We used the formula for the sum of an arithmetic series to find the sum of the first 21 terms of the given series. We also learned how to find the last term of an arithmetic series using the formula for the nth term. With this knowledge, you can now find the sum of any arithmetic series.

Frequently Asked Questions

Q: What is an arithmetic series?

A: An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant.

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: S_n = n/2 * (a + l), where S_n is the sum of the first n terms, n is the number of terms, a is the first term, and l is the last term.

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

A: To find the last term of an arithmetic series, you can use the formula: a_n = a + (n-1)d, where a_n is the nth term, a is the first term, n is the term number, and d is the common difference.

Q: What is the sum of the first 21 terms of the arithmetic series: -5, -3, -1, 1, ...?

In this article, we will continue to explore the concept of arithmetic series and answer some frequently asked questions.

Q: What is the difference between an arithmetic series and a geometric series?

A: An arithmetic series is a sequence of numbers in which the difference between any two consecutive terms is constant. A geometric series, on the other hand, is a sequence of numbers in which the ratio between any two consecutive terms is constant.

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

A: To find the sum of the first n terms of an arithmetic series, you can use the formula: S_n = n/2 * (a + l), where S_n is the sum of the first n terms, n is the number of terms, a is the first term, and l is the last term.

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

A: To find the last term of an arithmetic series, you can use the formula: a_n = a + (n-1)d, where a_n is the nth term, a is the first term, n is the term number, and d is the common difference.

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

A: The formula for the nth term of an arithmetic series is: a_n = a + (n-1)d, where a_n is the nth term, a is the first term, n is the term number, and d is the common difference.

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

A: To find the sum of an infinite arithmetic series, you can use the formula: S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio.

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

A: The formula for the sum of an infinite arithmetic series is: S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio.

Q: Can an arithmetic series have a negative common difference?

A: Yes, an arithmetic series can have a negative common difference. For example, the series: 5, 3, 1, -1, ... has a negative common difference of -2.

Q: Can an arithmetic series have a zero common difference?

A: Yes, an arithmetic series can have a zero common difference. For example, the series: 5, 5, 5, 5, ... has a zero common difference.

Q: What is the sum of the first 10 terms of the arithmetic series: 2, 5, 8, 11, ...?

A: To find the sum of the first 10 terms of the arithmetic series, we can use the formula: S_n = n/2 * (a + l), where S_n is the sum of the first n terms, n is the number of terms, a is the first term, and l is the last term.

First, we need to find the last term of the series. We can use the formula: a_n = a + (n-1)d, where a_n is the nth term, a is the first term, n is the term number, and d is the common difference.

In this case, the first term 'a' is 2, the common difference 'd' is 3, and the term number 'n' is 10. Plugging these values into the formula, we get:

a_10 = 2 + (10-1)3 = 2 + 9 * 3 = 2 + 27 = 29

So, the last term is 29.

Now, we can plug the values into the formula for the sum of an arithmetic series:

S_10 = 10/2 * (2 + 29) = 5 * 31 = 155

Therefore, the sum of the first 10 terms of the arithmetic series is 155.

Conclusion

In this article, we answered some frequently asked questions about arithmetic series and provided examples to illustrate the concepts. We also learned how to find the sum of an infinite arithmetic series and how to find the last term of an arithmetic series.

Frequently Asked Questions

Q: What is the difference between an arithmetic series and a geometric series?

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

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

A: To find the last term of an arithmetic series, you can use the formula: a_n = a + (n-1)d, where a_n is the nth term, a is the first term, n is the term number, and d is the common difference.

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

A: The formula for the nth term of an arithmetic series is: a_n = a + (n-1)d, where a_n is the nth term, a is the first term, n is the term number, and d is the common difference.

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

A: To find the sum of an infinite arithmetic series, you can use the formula: S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio.

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

A: The formula for the sum of an infinite arithmetic series is: S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio.