Geometric Sequence Calculator
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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Tap a sequence to load it.
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Geometric Sequence Calculator
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.
What Is a Geometric Sequence?
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
aₙ = a₁·r^(n−1) · Sₙ = a₁(1−rⁿ)/(1−r) · S∞ = a₁/(1−r), |r| < 1How to Use This Geometric Sequence Calculator
Enter the first term (a₁) of the sequence.
Enter the common ratio (r) — the number each term is multiplied by (use a fraction for decay, a negative for alternating).
Enter how many terms (n) you want.
Read the nth term, the sum of the first n terms, and the infinite sum (or 'diverges').
Review the listed terms, the progression graph, and the step-by-step formulas.
Common Use Cases
Math Homework
Solve nth-term, sum, and infinite-series problems with full working.
Learning the Formulas
See aₙ = a₁r^(n−1) and the sum formulas applied to real numbers.
Infinite Geometric Series
Find the sum of a converging series like 8 + 4 + 2 + … = 16.
Exponential Growth & Decay
Model doubling, halving, and percentage-based decay patterns.
Comparing to Arithmetic
See how multiplying (geometric) differs from adding (arithmetic) each term.
Exam Prep
Practice geometric-progression questions for algebra and pre-calc tests.
Why Use a Geometric Sequence Calculator?
Find Any Term Instantly
Get the 5th, 20th, or 100th term without multiplying through the whole sequence.
Finite and Infinite Sums
Compute the sum of the first n terms and, when it converges, the infinite sum S∞.
Convergence Made Clear
See immediately whether the series converges (|r| < 1) or diverges, with the value when it does.
Formulas Worked Out
The explicit and recursive formulas are shown with your numbers, not just the final answer.
Graph the Growth or Decay
A line chart shows the exponential pattern at a glance.
Frequently Asked Questions
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.