Geometric Sequence Calculator
Calculate the sum of a finite or infinite geometric series given a first term and common ratio. Useful in finance, physics, and any problem involving exponential growth or decay.
Last updated: May 2026
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About this calculator
A geometric sequence is one where each term is multiplied by a fixed common ratio r: aₙ = a × rⁿ⁻¹. The sum of the first n terms of a finite geometric series is: Sₙ = a × (rⁿ − 1) / (r − 1), valid when r ≠ 1. For an infinite geometric series, the sum converges only when |r| < 1, and the formula simplifies to: S∞ = a / (1 − r). If |r| ≥ 1 in an infinite series, the terms grow without bound and the series diverges — there is no finite sum. The ratio r can be any non-zero real number: r > 1 gives exponential growth, 0 < r < 1 gives decay, and negative r alternates sign each term. These formulas underpin compound interest, population models, and signal processing.
How to use
Suppose a = 2, r = 3, and you want the sum of the first 5 terms (finite series). Apply Sₙ = a × (rⁿ − 1) / (r − 1): S₅ = 2 × (3⁵ − 1) / (3 − 1) = 2 × (243 − 1) / 2 = 2 × 242 / 2 = 242. Now for an infinite series with a = 10, r = 0.5 (|r| < 1): S∞ = 10 / (1 − 0.5) = 10 / 0.5 = 20. Enter your values and choose finite or infinite to see your result.
Frequently asked questions
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 strictly less than 1, i.e., |r| < 1. In this case each successive term is smaller than the last, so the total approaches a limit given by S∞ = a / (1 − r). If |r| ≥ 1, the terms do not shrink and the series grows without bound — it diverges and has no finite sum. A classic example of convergence is the series 1 + 0.5 + 0.25 + … = 2.
What is the difference between a geometric sequence and a geometric series?
A geometric sequence is the list of individual terms: a, ar, ar², ar³, … A geometric series is the result of adding those terms together. The sequence 3, 6, 12, 24 becomes the series 3 + 6 + 12 + 24 = 45 when summed. Practically, you use the sequence to find a specific term at position n, while you use the series when you need the cumulative total after n steps.
How is the geometric series formula used in compound interest calculations?
Compound interest generates a geometric sequence of balances: each period the balance is multiplied by (1 + r), where r is the periodic interest rate. When you make equal periodic deposits, the total accumulated value is the sum of a geometric series. The formula Sₙ = a × (rⁿ − 1) / (r − 1) directly computes the future value of such an annuity. Understanding this connection shows why even small increases in interest rate or number of periods produce dramatically larger totals over time.