Skip to ContentGo to accessibility page
OpenStax Logo
Algebra 1

4.16.2 What Is an Arithmetic Sequence?

Algebra 14.16.2 What Is an Arithmetic Sequence?

4.16.2 What Is an Arithmetic Sequence?

4.16.2 • What Is an Arithmetic Sequence?

Activity

Here are the values of the first 5 terms of 3 sequences:

  1. 30, 40, 50, 60, 70, . . .
  2. 0, 5, 15, 30, 50, . . .
  3. 1, 2, 4, 8, 16, . . .

1. For each sequence, describe a way to produce a new term from the previous term.

a. 30, 40, 50, 60, 70, . . .

b. 0, 5, 15, 30, 50, . . .

c. 1, 2, 4, 8, 16, . . .

2. If the patterns you described continue, which sequence has the second-greatest value for the 10th10th term?

3. Which of these could be geometric sequences? Explain how you know.

An arithmetic sequence is a sequence in which each term is the previous term plus a constant. This constant is called the common difference.

4. Which of these could be arithmetic sequences? Explain how you know.

Are you ready for more?

Extending Your Thinking

Elena says that it’s not possible to have a sequence of numbers that is both arithmetic and geometric. Do you agree with Elena? Be prepared to show your reasoning.

Video: Comparing Sequences

Watch the following video to learn more about comparing arithmetic and geometric sequences.

Comparing Sequences

Self Check

Which of the following sequences is arithmetic?

  1. 25, 5, 1, 1 5 , . . .
  2. 3, 7, 11, 15, . . .
  3. 1, 4, 16, 64, . . .
  4. 3, 4, 6, 9, 13, . . .

Additional Resources

Arithmetic Sequences

Finding Common Differences

Companies often make large purchases, such as computers and vehicles, for business use.

  • A woman who starts a small contracting business purchases a new truck for $25,000.
  • After five years, she estimates that she will be able to sell the truck for $8,000. The loss in value of the truck will therefore be $17,000, which is $3,400 per year for five years.
  • The truck will be worth $21,600 after the first year; $18,200 after two years; $14,800 after three years; $11,400 after four years; and $8,000 at the end of five years.

The values of the truck in the example are said to form an arithmetic sequence because they change by a constant amount each year. Each term increases or decreases by the same constant value called the common difference of the sequence. For this sequence, the common difference is –3,400.

The sequence below is another example of an arithmetic sequence. In this case, the common difference is 3. You can choose any term of the sequence and add 3 to find the subsequent term.

ARITHMETIC SEQUENCE

An arithmetic sequence is a sequence that has the property that the difference between any two consecutive terms is a constant. This constant is called the common difference. If a 1 a 1 is the first term of an arithmetic sequence and d d is the common difference, the sequence will be:

{ a n } = { a 1 , a 1 + d , a 1 + 2 d , a 1 + 3 d . . . } { a n }={ a 1 , a 1 + d , a 1 + 2 d , a 1 + 3 d . . . }

Example

Is each sequence arithmetic? If so, find the common difference.

  1. {1, 2, 4, 8, 16, . . .}
  2. {3, 1, 5, 9, 13, . . .}

Subtract each term from the subsequent term to determine whether a common difference exists.

a. The sequence is not arithmetic because there is no common difference.

2 − 1 = 1 2 − 1 = 1       4 − 2 = 2 4 − 2 = 2       8 − 4 = 4 8 − 4 = 4       16 − 8 = 8 16 − 8 = 8

b. The sequence is arithmetic because there is a common difference. The common difference is 4.

1 − ( − 3 ) = 4 1 − ( − 3 ) = 4       5 − 1 = 4 5 − 1 = 4       9 − 5 = 4 9 − 5 = 4       13 − 9 = 4 13 − 9 = 4

Try it

Arithmetic Sequences

Is the given sequence arithmetic? If so, find the common difference.

{18, 16, 14, 12, 10, . . .}

PreviousNext
Order a print copy
Citation/Attribution
Reuse and redistribution of this content in digital or print format:
  • This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's prior written permission.
  • This book uses the Creative Commons Attribution-NonCommercial-ShareAlike License, which means that you can reuse and modify the material only for noncommercial purposes, must attribute OpenStax, and must distribute any derivative works under the same license.
  • Any commercial printing of this textbook, including using a local or custom printer, must be approved by OpenStax, and proper citation provided.
  • OpenStax-copyrighted images, activities, assessments, and similar components of this book are subject to the same licensing – CC-BY-NC-SA. They can be used for noncommercial purposes with attribution. Commercial use requires permission.
  • Permission requests: Anyone who intends to incorporate this content (including text, images, and other components) into large language models, use it in AI offerings, use it commercially (including in print), and/or has questions about another use case is welcome to complete our reuse request form.
Attribution information
  • If you are redistributing all or part of this book in a noncommercial print format, then you must include on every physical page the following attribution:

    Access for free at https://openstax.org/books/algebra-1/pages/about-this-course

  • If you are redistributing all or part of this book in a noncommercial digital format, then for every page that includes OpenStax content, you must license the derivative work under the same CC-BY-NC-SA license as the original, and include on every digital page view the following attribution:

    Access for free at https://openstax.org/books/algebra-1/pages/about-this-course

Citation information

The information below includes the information needed to generate citations in most major styles (APA, MLA, etc.); you must reformat and organize the information as needed to fit the requirements of the style. Use the information below to generate a citation. We recommend using a citation tool such as this one.

© Apr 23, 2026 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike License. The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo, and Rice University name, and Rice University logo trademarks, or wordmarks are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.