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A geometric sequence multiplies by the same fixed ratio at every step, which means it grows (or shrinks) far faster than an arithmetic sequence — and far faster than is practical to extend by hand. This calculator finds any term and the running sum of a geometric sequence directly from its first term and common ratio, with a full step-by-step solution.
How it works
Enter the first term, the common ratio between consecutive terms, and how many terms to generate. The calculator applies the nth term formula and the sum-of-n-terms formula directly.
- Enter first term (a₁).
- Enter common ratio (r).
- Enter number of terms (n).
- Click Calculate to see your results.
Examples
First term 3, ratio 2, 8 terms
The sequence runs 3, 6, 12, 24, ..., 384 — the 8th term is 384, and the sum of all 8 terms is 765.
Who should use it
- Modeling exponential growth or decay patterns like compound interest or population growth.
- Checking geometric sequence homework problems for algebra or precalculus courses.
Industry applications
- Algebra and precalculus education
- Finance (compound growth modeling) and computer science (algorithmic complexity analysis)
Advantages
- Calculates the nth term, the sum of n terms, and lists every term in the sequence.
- Correctly handles the special case where the common ratio equals 1.
Limitations
- Limited to listing up to 100 terms at a time — geometric sequences can grow astronomically large well before that limit.
Common mistakes to avoid
- Confusing a geometric sequence (constant ratio) with an arithmetic sequence (constant difference) — they use completely different formulas.
- Forgetting that a negative common ratio makes the sequence alternate between positive and negative terms.
Best practices
- Double-check the common ratio by dividing two consecutive known terms before relying on a calculated nth term far into the sequence.
Tips
- Geometric sequences model compound growth directly — a common ratio of 1.05, for example, represents 5% growth per period, the same math behind compound interest.