How to Find a Number of Terms in an Arithmetic Sequence

Co-authored by Sophia Bell Reviewed by Grace Imson, MA

Last Updated: September 19, 2024 Fact Checked

Finding the number of terms in an arithmetic sequence might sound like a complex task, but it’s actually pretty straightforward. All you need to do is plug the given values into the formula t_n = a + (n - 1) d and solve for n, which is the number of terms. Note that t_n is the last number in the sequence, a is the first term in the sequence, and d is the common difference. ^{[1]
X
Research source}

Steps

Download Article

1
Identify the first, second, and last terms of the sequence. Typically, to solve a problem like this, you’ll be given the first 3 or more terms as well as the last term.^{[2]
X
Research source}
- For example, you may have the following sequence: 107, 101, 95…-61. In this case, the first term is 107, the second term is 101, and the last term is -61. You need all of this information to solve the problem.
In the example sequence, the first term is 107 and the second term is 101. So, subtract 107 from 101, which is -6. Therefore, the common difference is -6.^{[3]
X
Research source}

Advertisement
3
Use the formula t_n = a + (n - 1) d to solve for n. Plug in the last term (t_n), the first term (a), and the common difference (d).^{[4]
X
Research source} Work through the equation until you’ve solved for n.^{[5]
X
Research source}
- For example, start by writing: -61 = 107 + (n - 1) -6. Subtract 107 from both sides so you’re left with -168 = (n - 1) -6. Then, divide both sides by -6 to get 28 = n - 1. Finish by adding 1 to both sides so that n = 29.^{[6]
  X
  Research source}

Community Q&A

Search

Add New Question

Question

Can I use the formula (A - L/d) + 1 for finding n?

Donagan

Top Answerer

As explained above, n = [(L - A) / d] + 1.

Thanks! We're glad this was helpful.
Thank you for your feedback.
If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow
Yes No

Not Helpful 51 Helpful 79
Question

If the difference is not given, how do I find the difference?

Donagan

Top Answerer

Subtract any term from the one that follows it.

Thanks! We're glad this was helpful.
Thank you for your feedback.
If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow
Yes No

Not Helpful 68 Helpful 65
Question

If the first term and last term of an arithmetic progression are 5 and 89, how do I find the number of terms?

Donagan

Top Answerer

You can't without knowing the difference between consecutive terms.

Thanks! We're glad this was helpful.
Thank you for your feedback.
If wikiHow has helped you, please consider a small contribution to support us in helping more readers like you. We’re committed to providing the world with free how-to resources, and even $1 helps us in our mission. Support wikiHow
Yes No

Not Helpful 88 Helpful 105