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Geometric Series Sum Calculator

A geometric progression (GP) is a sequence where each term equals the previous one multiplied by a constant ratio r. Enter first term a, common ratio r and number of terms n to instantly get the nth term aₙ, the partial sum Sₙ, the mean, the infinite sum (when |r| < 1) and a preview of the first 30 terms. Widely used in school maths (GCSE / A-Level / HKDSE), compound interest derivations, exponential population growth, half-life problems and Zeno-style paradoxes.

nth term aₙ

512

Sum of first n terms Sₙ

1,023

Mean a̅

102.3

Infinite sum S∞

—

|r| ≥ 1 — the infinite series diverges.

First 30 terms

At most the first 30 terms are listed — extend with the formula.

    aₙ = a · r^(n − 1) Sₙ = a · (1 − rⁿ) / (1 − r) S∞ = a / (1 − r) when |r| < 1

    aₙ = a · r^(n−1)   Sₙ = a · (1 − rⁿ) / (1 − r)   S∞ = a / (1 − r) [|r| < 1]

    Each term is the previous one multiplied by a fixed common ratio r. When |r| < 1 the infinite sum converges to a / (1 − r); otherwise it diverges.

    Formula

    aₙ = a · r^(n − 1) Sₙ = a · (1 − rⁿ) / (1 − r) when r ≠ 1 Sₙ = n · a when r = 1 S∞ = a / (1 − r) when |r| < 1

    Frequently asked

    How is a geometric series different from an arithmetic series?

    In an arithmetic progression each term adds a fixed amount d (e.g. 2, 5, 8, 11 with d = +3). In a geometric progression each term multiplies by a fixed ratio r (e.g. 2, 6, 18, 54 with r = 3). Arithmetic captures linear growth (a fixed monthly raise); geometric captures exponential growth (compound interest, population doubling, radioactive decay). Sum formulas differ: Sₙ = n/2 · (2a + (n − 1)d) for arithmetic; Sₙ = a · (1 − rⁿ) / (1 − r) for geometric.

    When does the infinite geometric sum converge?

    The infinite sum converges if and only if the absolute value of the common ratio is less than 1 (|r| < 1, i.e. −1 < r < 1). The terms then shrink toward zero and Sₙ approaches the limit S∞ = a / (1 − r). Classic example: 1 + 1/2 + 1/4 + 1/8 + … = 2 (a = 1, r = 0.5). When |r| ≥ 1 the terms do not shrink (or grow unboundedly) and the sum diverges, so no finite total exists.

    Why does r = 1 need a different formula?

    The general formula Sₙ = a · (1 − rⁿ) / (1 − r) has 1 − r in its denominator. When r = 1 that denominator becomes zero, so the formula is undefined. But when r = 1 the sequence is just a, a, a, … (a constant sequence) and its partial sum is obviously Sₙ = n · a. This calculator handles that edge case automatically.

    Can I use this to compute compound interest or annuities?

    You can use it to understand the underlying maths, but a dedicated tool is faster. Compound interest is the nth term of a GP (each period multiplies by 1 + i), and a fixed-payment annuity total is the sum of a GP. For those use cases, see this site's dedicated Compound Interest, Annuity Payment and Present Value calculators.

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