This geometric sequence calculator finds the nth term (aₙ = a₁·rⁿ⁻¹), the sum of the first n terms, and — when |r| < 1 — the infinite series sum.

nth term aₙ –
Sum of first n terms Sₙ –
Infinite sum S∞ –

The Constant-Multiple Sequence: aₙ = a₁ · rⁿ⁻¹

Its additive cousin is our Arithmetic Sequence Calculator. Part of our Math Calculators collection.

Worked Examples (Verified)

  • a₁ = 5, r = 0.5, n = 10 (defaults): a₁₀ = 5 × 0.5⁹ = 0.00977 · S₁₀ = 5 × (1 − 0.5¹⁰)/0.5 = 9.99 · S∞ = 5/0.5 = 10 — the finite sum chasing its limit and nearly kissing it
  • Doubling (r = 2): a₁ = 1 gives 1, 2, 4, 8… and Sₙ = 2ⁿ − 1 — the chessboard-and-rice legend is this row with kingdoms at stake
  • r between −1 and 0 alternates signs while shrinking: 8, −4, 2, −1… converging to 8/1.5 ≈ 5.33 — convergence doesn't require politeness, only |r| < 1

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Frequently Asked Questions

What is the formula for a geometric sequence?

aₙ = a₁ · rⁿ⁻¹, with a₁ the first term and r the common ratio (any term divided by its predecessor). Sum of n terms: Sₙ = a₁(1 − rⁿ)/(1 − r) for r ≠ 1. This calculator evaluates both, plus the infinite sum when it exists.

When does an infinite geometric series have a sum?

Exactly when |r| < 1, and then S∞ = a₁/(1 − r). The defaults: 5/(1 − 0.5) = 10. At |r| ≥ 1 the terms refuse to shrink toward zero, so partial sums never settle — the series diverges.

How do you find the common ratio?

Divide successive terms: r = a₂/a₁ = a₃/a₂ (any adjacent pair). From distant terms, r = (aₙ/aₘ)^(1/(n−m)) — with the sign ambiguity resolved by term signs when the gap is even.

Is compound interest a geometric sequence?

Yes — yearly balances with rate i form a geometric sequence with r = 1 + i. That's why this same mathematics prices annuities, models populations and decays radioactive samples (r < 1): one formula, every exponential story.

What does r = 1 or r = −1 do?

r = 1: the constant sequence, Sₙ = n·a₁ (the sum formula's denominator vanishes, hence the special case). r = −1: terms alternate a₁, −a₁, a₁… and partial sums oscillate without limit — divergent by oscillation.

Is this geometric sequence calculator free?

Yes — free, instant, no sign-up.