Geometric Sequence Sum Calculator
A geometric sequence multiplies each term by a fixed common ratio. Given the first term, the common ratio, and the number of terms, this calculator returns the nth term and the sum of those terms. It also reports the sum to infinity when the series converges (when the absolute value of the ratio is below 1). Geometric sums underpin compound interest, loan amortisation, and the geometry of repeated scaling.
Geometric sequence sum formula
nth term: a_n = a * r^(n - 1)
Sum (r != 1): S = a * (1 - r^n) / (1 - r)
Sum (r = 1): S = n * a
Sum to infinity (|r| < 1): S = a / (1 - r)
The infinite sum is shown only when the absolute value of r is less than 1; otherwise it does not converge.
Worked example
With a = 2, r = 3, n = 5: the nth term is 2 * 3^4 = 2 * 81 = 162. The sum is 2 * (1 - 3^5) / (1 - 3) = 2 * (1 - 243) / (-2) = 2 * (-242)/(-2) = 242. The series does not converge because the ratio is 3.
Geometric sequence sum: frequently asked questions
What is a geometric sequence?
A geometric sequence multiplies each term by a fixed common ratio r to get the next term. For example 2, 6, 18, 54 has first term 2 and common ratio 3.
What is the formula for the sum of a geometric sequence?
When the common ratio r is not 1, the sum of the first n terms is S = a times (1 minus r to the power n) divided by (1 minus r). When r equals 1 every term is the same, so the sum is simply n times a.
When does an infinite geometric series have a finite sum?
An infinite geometric series converges to a finite sum only when the absolute value of the common ratio is less than 1. In that case the sum to infinity is a divided by (1 minus r). If the absolute value of r is 1 or more, the infinite series does not converge.
Can the common ratio be negative?
Yes. A negative ratio produces an alternating sequence whose terms switch sign. For example a = 1, r = -2 gives 1, -2, 4, -8. The partial-sum formula still applies.
Sources
- NIST Digital Library of Mathematical Functions: dlmf.nist.gov (series reference).
- Standard geometric series partial-sum and convergence formulas.
Reviewed by the CalculatorHub team, edited by James Graham, 19 June 2026. See our methodology.