How this arithmetic sequence calculator works
An arithmetic sequence is a list of numbers in which the difference between each term and the next is always the same. That constant gap is the common difference, written d. Starting from the first term a₁, you add d again and again: a₁, a₁+d, a₁+2d, a₁+3d, and so on.
To find any term directly, use the nth term formula aₙ = a₁ + (n−1)d — you add the common difference (n−1) times to the first term. To add up the first n terms, use the sum formula Sₙ = n/2 × (2a₁ + (n−1)d), which is the same as Sₙ = n(a₁ + aₙ)/2: the number of terms multiplied by the average of the first and last term. This calculator computes both, lists the terms (capped at 100 for readability), and can also work backwards to find d from two terms or the position n of a known value.
Reference — common difference and term examples:
| a₁ | d | Sequence start | a₅ | S₅ |
|---|---|---|---|---|
| 3 | 5 | 3, 8, 13, 18, 23 | 23 | 65 |
| 2 | 2 | 2, 4, 6, 8, 10 | 10 | 30 |
| 10 | −3 | 10, 7, 4, 1, −2 | −2 | 20 |
| 0 | 1 | 0, 1, 2, 3, 4 | 4 | 10 |
Frequently asked questions
- What is an arithmetic sequence?
- An arithmetic sequence is a list of numbers where the gap between consecutive terms is always the same. That fixed gap is the common difference, d. For example 3, 8, 13, 18, 23 is arithmetic because each term is 5 more than the one before it.
- What is the nth term formula for an arithmetic sequence?
- The nth term is aₙ = a₁ + (n−1)d, where a₁ is the first term, d is the common difference and n is the position. You add the common difference (n−1) times to the first term. With a₁ = 3 and d = 5, the 10th term is 3 + (10−1)×5 = 48.
- How do I find the sum of an arithmetic sequence?
- The sum of the first n terms is Sₙ = n/2 × (2a₁ + (n−1)d), equivalently Sₙ = n(a₁ + aₙ)/2 — the number of terms times the average of the first and last term. For a₁ = 3, d = 5, n = 10, the last term is 48, so Sₙ = 10 × (3 + 48)/2 = 255.
- What is the common difference?
- The common difference, d, is the constant amount added to each term to get the next term. It can be positive (increasing), negative (decreasing) or zero (constant). You find it by subtracting any term from the term that follows it.
- What is the difference between an arithmetic and a geometric sequence?
- In an arithmetic sequence you add a fixed common difference d, so terms grow linearly (3, 8, 13, 18…). In a geometric sequence you multiply by a fixed common ratio r, so terms grow multiplicatively (3, 6, 12, 24…).
- How do I find the common difference from two terms?
- If you know two terms and their positions, d = (aₘ − aₖ)/(m − k): the change in value divided by the change in position. For example if the 4th term is 18 and the 7th term is 33, then d = (33 − 18)/(7 − 4) = 5.