Geometric Sequence Calculator
nth term, partial sum and infinite sum of a geometric sequence from the first term and ratio.
In short
nth term, partial sum and infinite sum of a geometric sequence from the first term and ratio.
Formula: aₙ = a₁ × r^(n−1); Sₙ = a₁(1 − rⁿ) ÷ (1 − r) for r ≠ 1 (Sₙ = n a₁ when r = 1); infinite sum = a₁ ÷ (1 − r) when |r| < 1.
What this calculator does
nth term, partial sum and infinite sum of a geometric sequence from the first term and ratio. Formula, worked example, sources, FAQs and limitations included.
You enter first term a₁, common ratio r and term number n, and the geometric sequence calculator shows the answer along with the working, so you can check it or try other numbers.
The page shows the formula, a numeric worked example, and the assumptions that affect this math calculation.
Inputs and what they mean
- First term a₁
- — number value.
- Common ratio r
- — number value.
- Term number n
- — number value.
How to use it
- Enter each value in the unit shown next to the box (use the unit converter first if your numbers are in other units).
- Check the breakdown to see every intermediate step.
- Read the limitations before relying on the result.
Formula
Source: OpenStax College Algebra 2e §9.3 Geometric Sequences and §9.4 Series.
aₙ = a₁ × r^(n−1); Sₙ = a₁(1 − rⁿ) ÷ (1 − r) for r ≠ 1 (Sₙ = n a₁ when r = 1); infinite sum = a₁ ÷ (1 − r) when |r| < 1.
Inputs used: First term a₁, Common ratio r, Term number n.
Worked example
2, 6, 18, … to the 5th term
- a₅ = 2 × 3⁴ = 162.
- S₅ = 2(1 − 243) ÷ (1 − 3) = 242.
Reading the result
The headline figure is the main answer. Any breakdown underneath shows the parts that make it up, so you can check the working and see what changes when you adjust an input.
Limitations and assumptions
- Very large n or r can overflow; the calculator then reports an out-of-range result.
- Floating-point rounding applies.
- Exact mathematical relationships, rounded for display. Check units and rounding before using a result in graded, engineering or financial work.
- The result depends on the values you enter for this geometric sequence calculator; it does not supply missing rates, rules, prices, dates, or assumptions for you.
Reference: OpenStax
Last reviewed:
Common questions
When does an infinite geometric series converge?
Only when |r| < 1; the sum is a₁ ÷ (1 − r).
Can r be negative?
Yes. Terms then alternate in sign.
How do I find r?
Divide any term by the previous one.
How do I use the Geometric Sequence Calculator?
Enter the required values for First term a₁, Common ratio r, Term number n. The calculator applies the formula on this page and shows the main result with any supporting breakdown so you can check the arithmetic.
What formula does the Geometric Sequence Calculator use?
aₙ = a₁ × r^(n−1); Sₙ = a₁(1 − rⁿ) ÷ (1 − r) for r ≠ 1 (Sₙ = n a₁ when r = 1); infinite sum = a₁ ÷ (1 − r) when |r| < 1. The visible formula section above lists the calculation path and the edge cases the page handles, so the result can be checked without relying on the form alone.
Can the Geometric Sequence Calculator be used for exact decisions?
Use it as a calculation aid, not as a substitute for checking the underlying rule, contract, policy, or professional advice that applies to your situation. When a result depends on local rules, personal details, prices, or dates, enter those values yourself and confirm them before acting.
Sources
Method based on OpenStax.
Cite this page
Free Calculator. (2026, November 3). Geometric Sequence Calculator. https://freecalculator.tech/tools/geometric-sequence-calculator
"Geometric Sequence Calculator." Free Calculator, 3 Nov. 2026, https://freecalculator.tech/tools/geometric-sequence-calculator.
Geometric Sequence Calculator — Free Calculator, https://freecalculator.tech/tools/geometric-sequence-calculator
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