Arithmetic Series Sum Calculator
An arithmetic series is the sum of the terms of an arithmetic sequence: a sequence where the difference between consecutive terms is constant. The first term is a, the common difference is d, and the number of terms is n. The sum formula S = n/2 * (2a + (n-1)d) was derived by noticing that pairing the first and last terms, second and second-to-last terms, and so on, always gives the same total. This insight, popularized by Gauss, extends to any finite arithmetic sequence. Enter the three parameters below to compute the sum and the value of the last (nth) term.
Arithmetic series formula
S = n/2 * (2a + (n-1)d)
Last term L = a + (n-1)d
Equivalent form: S = n * (a + L) / 2
Where a is the first term, d is the common difference, and n is the number of terms. The formula works for positive, negative, or zero common differences and for any real first term.
Arithmetic series properties
- When d = 0, all terms are equal to a, and S = n * a.
- When a = 1 and d = 1, the formula gives S = n(n+1)/2, the classic triangular number formula.
- The average (mean) of the terms equals (a + L)/2, the midpoint of the sequence.
- Arithmetic sequences have a linear relationship between term index and value.
- The sum of the first n odd numbers (a = 1, d = 2) is always n^2.
Arithmetic series: frequently asked questions
What is an arithmetic series?
An arithmetic series is the sum of the terms of an arithmetic sequence, where each term increases by a fixed common difference d. For example, 2 + 5 + 8 + 11 + 14 is an arithmetic series with first term a = 2, d = 3, and 5 terms.
What is the formula for the sum of an arithmetic series?
S = n/2 * (2a + (n-1)d), where n is the number of terms, a is the first term, and d is the common difference. An equivalent form is S = n * (a + L)/2, where L is the last term, because the series is symmetric around its midpoint.
How do I find the nth term of an arithmetic sequence?
The nth term is a(n) = a + (n-1)d. For example, in the sequence 3, 7, 11, 15, ..., the 10th term is 3 + (10-1)*4 = 3 + 36 = 39.
Who derived the arithmetic series formula?
The formula is attributed to Gauss, who as a schoolboy famously computed the sum 1 + 2 + ... + 100 = 5,050 by pairing the first and last terms (1 + 100 = 101) and multiplying by the number of pairs (50). The general formula follows from the same pairing argument.
What if the common difference is negative?
A negative common difference produces a decreasing sequence. The sum formula works equally well: S = n/2 * (2a + (n-1)d). For example, 10 + 7 + 4 + 1 has a = 10, d = -3, n = 4, giving S = 4/2 * (20 + 3*(-3)) = 2 * 11 = 22.
Official sources
- NIST Digital Library of Mathematical Functions: dlmf.nist.gov.
- NIST, Mathematical topics: nist.gov/topics/mathematics.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.