Find the nth term, sum of n terms, and infinite sum of a geometric sequence from the first term, common ratio, and count — with formula and steps.
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A geometric sequence is a list of numbers where each term is found by multiplying the previous one by a fixed number called the common ratio. This calculator finds any term, the sum of the first n terms, and — when the sequence converges — the infinite sum, all from three inputs: the first term, the common ratio, and how many terms you want. It shows the formulas with your numbers plugged in, lists the terms, graphs the progression, and works step by step, so it doubles as a study aid.
A geometric sequence (or geometric progression) is a sequence in which each term equals the previous term multiplied by a constant, the common ratio r. Starting from the first term a₁, the terms are a₁, a₁r, a₁r², and so on. The nth term is aₙ = a₁ · r^(n−1), and the sum of the first n terms is Sₙ = a₁(1 − rⁿ)/(1 − r). If the common ratio is between −1 and 1, the sequence shrinks toward zero and the infinite sum converges to S∞ = a₁/(1 − r); otherwise the series diverges and has no finite total. Unlike an arithmetic sequence, which adds a constant each step, a geometric sequence multiplies — producing exponential growth or decay.
nth Term, Sum & Infinite Sum
Solve nth-term, sum, and infinite-series problems with full working.
See aₙ = a₁r^(n−1) and the sum formulas applied to real numbers.
Find the sum of a converging series like 8 + 4 + 2 + … = 16.
Model doubling, halving, and percentage-based decay patterns.
See how multiplying (geometric) differs from adding (arithmetic) each term.
Practice geometric-progression questions for algebra and pre-calc tests.
Get the 5th, 20th, or 100th term without multiplying through the whole sequence.
Compute the sum of the first n terms and, when it converges, the infinite sum S∞.
See immediately whether the series converges (|r| < 1) or diverges, with the value when it does.
The explicit and recursive formulas are shown with your numbers, not just the final answer.
A line chart shows the exponential pattern at a glance.
The nth term is aₙ = a₁ · r^(n−1), where a₁ is the first term, r is the common ratio, and n is the term number. For example, with a₁ = 3 and r = 2, the 5th term is 3 · 2⁴ = 48.
For the first n terms, use Sₙ = a₁(1 − rⁿ)/(1 − r) when r ≠ 1. For example, 3 + 6 + 12 + 24 + 48 has sum 3(1 − 2⁵)/(1 − 2) = 93. If r = 1, every term equals a₁, so the sum is simply n × a₁.
An infinite geometric series converges only when the absolute value of the common ratio is less than 1 (|r| < 1). Then the infinite sum is S∞ = a₁/(1 − r). For example, 8 + 4 + 2 + 1 + … converges to 8/(1 − 0.5) = 16. If |r| ≥ 1, the series diverges and has no finite sum.
The common ratio, r, is the constant each term is multiplied by to get the next term. Find it by dividing any term by the one before it: r = a₂/a₁. A ratio above 1 grows, between 0 and 1 shrinks, and negative ratios alternate sign.
An arithmetic sequence adds a constant (the common difference) each term, giving a straight-line pattern. A geometric sequence multiplies by a constant (the common ratio) each term, giving exponential growth or decay. Use the arithmetic sequence calculator for the additive case.
This is geometric with a₁ = 3 and r = 2, so the 5th term is a₅ = 3 · 2⁴ = 3 · 16 = 48. The sum of the first 5 terms is 93.