Sum of Series Calculator
A series is the sum of the terms of a sequence. This calculator sums two of the most common kinds: arithmetic, where each term increases by a fixed difference, and geometric, where each term is multiplied by a fixed ratio. Enter the first term, the common difference or ratio, and how many terms to add, and it returns the total sum and the value of the last term using the standard closed-form formulas. It is built for algebra, finance, and any setting where you need a quick, exact running total of a regular sequence.
Series sum formulas
Arithmetic sum = n/2 * (2a + (n - 1) * d)
Arithmetic nth term = a + (n - 1) * d
Geometric sum (r != 1) = a * (1 - r^n) / (1 - r)
Geometric sum (r = 1) = a * n
Geometric nth term = a * r^(n - 1)
These closed forms give the exact total without adding terms one by one. The term count n must be a positive whole number.
Series context
- Arithmetic series have a constant gap between terms; geometric series have a constant multiplier.
- The sum of the first n whole numbers (a = 1, d = 1) is n(n + 1) / 2.
- Geometric sums model compound growth such as repeated percentage increases.
- An infinite geometric series converges only when the ratio's magnitude is below 1.
- A term count below 1 returns n/a.
Sum of series: frequently asked questions
What is an arithmetic series?
An arithmetic series adds terms that increase by a fixed common difference. With first term a, common difference d, and n terms, the sum equals n divided by 2 times the quantity (2a plus (n minus 1) times d). The nth term is a plus (n minus 1) times d.
What is a geometric series?
A geometric series multiplies each term by a fixed common ratio. With first term a, common ratio r (not equal to 1), and n terms, the sum equals a times (1 minus r to the power n) divided by (1 minus r). When r equals 1 the sum is simply a times n.
Which series type should I pick?
Choose arithmetic when each term differs from the last by a constant amount (1, 4, 7, 10). Choose geometric when each term is a constant multiple of the last (2, 6, 18, 54). The calculator applies the matching closed-form sum.
Can the number of terms be a fraction?
No. The number of terms must be a positive whole number, because a series sums a discrete count of terms. The calculator rounds the term count and requires it to be at least 1.
Does a geometric series converge?
An infinite geometric series converges only when the absolute value of the common ratio is less than 1. This calculator sums a finite number of terms, so it always returns a value, but for fewer than infinite terms the ratio can be any value.
Official sources
- NIST Digital Library of Mathematical Functions: DLMF home.
- NIST: National Institute of Standards and Technology.
Reviewed by the CalculatorHub team, edited by James Graham, 17 June 2026. See our methodology.