Math · Sequences
Arithmetic sequence calculator.
Give the first term and common difference, or any two terms. You get the nth term, the sum of the first n terms, the first few terms and the steps.
For a first term of 3, a common difference of 4 and n = 10, the 10th term is 3 + 9 × 4 = 39 and the sum of the first 10 terms is 10 × (3 + 39) / 2 = 210. Checking 43: it is term 11.
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The formulas
An arithmetic sequence adds the same common difference d each step. The nth term is aₙ = a₁ + (n − 1)d. The sum of the first n terms is Sₙ = n(a₁ + aₙ) / 2, which equals n/2 × (2a₁ + (n − 1)d). If you know two terms aₚ and a_q, then d = (a_q − aₚ) / (q − p) and a₁ = aₚ − (p − 1)d.
Worked example
a₁ = 3, d = 4, n = 10: a₁₀ = 3 + 9 × 4 = 39 and S₁₀ = 10 × (3 + 39) / 2 = 210. The sequence starts 3, 7, 11, 15, 19, 23, 27, 31, 35, 39. To test whether 43 is a term, n = (43 − 3) / 4 + 1 = 11, a whole number, so it is the 11th term. From two terms, a₃ = 11 and a₇ = 27 give d = (27 − 11) / 4 = 4 and a₁ = 11 − 2 × 4 = 3, so a₂₀ = 79 and S₂₀ = 820.
Limits
The common difference can be negative or zero. n must be a whole number. Very large values are shown in exponent form.
Sources, opened 15 October 2026: OpenStax, Algebra and Trigonometry 2e, 13.2 Arithmetic Sequences (common difference; explicit formula aₙ = a₁ + d(n − 1)); OpenStax, Algebra and Trigonometry 2e, 13.4 Series and Their Notations (sum of the first n terms of an arithmetic series, Sₙ = n(a₁ + aₙ)/2).
Frequently asked questions
How do you find the nth term of an arithmetic sequence?
Use aₙ = a₁ + (n − 1)d. With a₁ = 3 and d = 4, the 10th term is 3 + 9 × 4 = 39.
How do you find the sum of an arithmetic sequence?
Sₙ = n(a₁ + aₙ) / 2. For 3, 7, 11, ... up to the 10th term 39, the sum is 10 × (3 + 39) / 2 = 210.
How do I find the common difference from two terms?
Divide the difference of the term values by the difference of their positions: d = (a_q − aₚ) / (q − p). For a₃ = 11 and a₇ = 27, d = 16 / 4 = 4.
How can I tell if a number is in the sequence?
Solve n = (value − a₁) / d + 1. If n is a whole number of 1 or more, the value is term n. With a₁ = 3 and d = 4, the value 43 gives n = 11, so it is the 11th term.
What is the difference between a sequence and a series?
A sequence is the list of terms. A series is the sum of the terms, so Sₙ is the arithmetic series of the first n terms.
Learning aid
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