Geometric Sequence Calculator

The formulas

In a geometric sequence every term equals the previous term times the common ratio r. Starting from the first term a₁:

aₙ = a₁ · r^(n−1)

The sum of the first n terms (the partial sum) is:

Sₙ = a₁ · (1 − rⁿ) / (1 − r) for r ≠ 1

When the ratio is exactly 1 every term is the same, so the sum is simply Sₙ = n · a₁. If the ratio lies strictly between −1 and 1, the infinite sum converges to a₁ / (1 − r).

Worked example

Take a₁ = 2 and r = 3, and ask for n = 5 terms.

• Sequence: 2, 6, 18, 54, 162
• 5th term: a₅ = 2 · 3⁴ = 2 · 81 = 162
• Sum: S₅ = 2 · (1 − 3⁵) / (1 − 3) = 2 · (1 − 243) / (−2) = 242

Each term is exactly 3× the one before it, and the five terms add up to 242.

Terms and partial sums at a glance

Common pitfalls

• Off-by-one on the exponent. The nth term uses r^(n−1), not rⁿ, the first term has no ratio applied yet.
• Ratio of 1. Plug r = 1 into the main sum formula and you divide by zero. Use Sₙ = n · a₁ instead; this tool switches automatically.
• Geometric vs. arithmetic. A geometric sequence multiplies by a fixed ratio; an arithmetic one adds a fixed difference. Mixing them up gives wildly wrong terms.
• Negative ratios. A ratio like −2 alternates the sign of the terms (2, −4, 8, −16, …); that is valid and the calculator handles it.

Where geometric sequences show up

Geometric sequences model anything that grows or shrinks by a constant percentage each step. Compound interest multiplies a balance by the same factor every period, so the balances form a geometric sequence. Population growth, radioactive decay measured in half-lives, the bounce heights of a dropped ball, and the depreciation of an asset each year all follow the same pattern. Recognising the common ratio in real data lets you project the next value, estimate a running total, or judge whether the convergent infinite-sum formula applies when the ratio sits between −1 and 1.