Arithmetic Sequence Calculator
An arithmetic sequence goes up or down by the same amount each step. Enter the first term, the common difference and a term number to get that term, the sum of all the terms up to it, the rule for the nth term and the opening terms of the sequence.
- Sum of the first n terms (Sₙ)
- 155
- Sequence
- 2, 5, 8, 11, 14, 17, 20, 23, 26, 29
- Rule for the nth term
- aₙ = 3n - 1
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Formula
How to use it
- Enter the first term.
- Enter the common difference — the amount added each step; negative if the sequence decreases.
- Enter which term you want, n.
- Read the nth term and the sum of the first n terms.
Worked examples
2, 5, 8, 11 … the 10th term
- nth term (aₙ)
- 29
- Sum of the first n terms (Sₙ)
- 155
- Sequence
- 2, 5, 8, 11, 14, 17, 20, 23, 26, 29
- Rule for the nth term
- aₙ = 3n - 1
Adding the whole numbers from 1 to 100
- nth term (aₙ)
- 100
- Sum of the first n terms (Sₙ)
- 5,050
50, 46, 42 … the 12th term
- nth term (aₙ)
- 6
- Sum of the first n terms (Sₙ)
- 336
- Rule for the nth term
- aₙ = -4n + 54
Finding the common difference
Subtract any term from the one after it: in 2, 5, 8, 11 the difference is 3. If you know two terms that are not neighbors, divide the gap in value by the gap in position: a 3rd term of 11 and a 7th term of 27 give d = (27 − 11) ÷ (7 − 3) = 4, so the first term is 3.
Why the sum formula works
Pair the first term with the last, the second with the second-to-last, and so on: every pair has the same total. Adding 1 to 100 makes 50 pairs that each sum to 101, so the total is 5,050 — the trick attributed to the young Gauss.
Questions people ask
What is the 10th term of 2, 5, 8, 11…?
29. With a first term of 2 and a difference of 3, a₁₀ = 2 + 9 × 3.
What is the sum of the whole numbers from 1 to 100?
5,050, from 100 × (1 + 100) ÷ 2.
What is the difference between an arithmetic and a geometric sequence?
An arithmetic sequence adds a fixed amount each step (2, 5, 8, 11); a geometric sequence multiplies by a fixed ratio (3, 6, 12, 24).