Arithmetic Sequence Sum Calculator
An arithmetic sequence adds a fixed common difference between each term. Given the first term, the common difference, and how many terms you want, this calculator returns the last term, the total sum of those terms, and the average term. The sum formula is one of the oldest results in mathematics, attributed to a young Gauss adding the numbers 1 through 100.
Arithmetic sequence sum formula
nth term: a_n = a + (n - 1) * d
Sum: S = n / 2 * (2a + (n - 1) * d)
Equivalently: S = n * (a + a_n) / 2
The sum equals the number of terms times the average of the first and last terms. Both forms give the same result.
Worked example
With a = 3, d = 4, n = 10: the nth term is 3 + 9*4 = 39. The sum is 10/2 * (3 + 39) = 5 * 42 = 210. The average term is 210/10 = 21.
Arithmetic sequence sum: frequently asked questions
What is an arithmetic sequence?
An arithmetic sequence is a list of numbers in which each term is obtained by adding a fixed value, the common difference d, to the previous term. For example 3, 7, 11, 15 has first term 3 and common difference 4.
What is the formula for the sum of an arithmetic sequence?
The sum of the first n terms is S = n divided by 2, times (2 times the first term a, plus (n minus 1) times the common difference d). Equivalently S = n times the average of the first and last terms.
How do I find the nth term?
The nth term is the first term plus (n minus 1) times the common difference: a_n = a + (n - 1) times d. This calculator returns the last (nth) term alongside the total sum.
Can the common difference be negative?
Yes. A negative common difference produces a decreasing sequence. For example a = 20, d = -3 gives 20, 17, 14 and so on. The sum formula works the same way with a negative d.
Sources
- NIST Digital Library of Mathematical Functions: dlmf.nist.gov (sequences and series reference).
- Standard arithmetic series partial-sum formula.
Reviewed by the CalculatorHub team, edited by James Graham, 19 June 2026. See our methodology.