Math
Geometric Series Calculator
A geometric series adds terms that each multiply the last by the same ratio r, like 2 + 6 + 18 + 54.
Enter the first term, ratio, and number of terms for the finite sum, or choose the infinite sum and the calculator checks |r| < 1 before applying S = a / (1 - r).
Negative ratios alternate signs. |r| < 1 shrinks.
Try an example
Result
Sum
—
- Last term
- —
- Infinite sum
- —
- Partial sums
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- Tail beyond n terms
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Student quick launch
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Study path
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What a geometric series is
Each term is the previous term times a constant ratio r, so the series is a + ar + ar² + ... + arⁿ⁻¹. Doubling patterns (r = 2), halving patterns (r = 1/2), and compound interest (r = 1.05) are all geometric.
Why the formula works
Multiply the sum by r and subtract: Sₙ - rSₙ = a - arⁿ, because every middle term cancels. Factor out Sₙ and divide by (1 - r). If r = 1 nothing cancels and the sum is just n copies of a.
The infinite case
| Series | a | r | Sum |
|---|---|---|---|
| 1 + 1/2 + 1/4 + ... | 1 | 1/2 | 2 |
| 0.9 + 0.09 + 0.009 + ... | 0.9 | 0.1 | 1 (so 0.999... = 1) |
| 1 - 1/3 + 1/9 - ... | 1 | -1/3 | 3/4 |
| 1 + 2 + 4 + ... | 1 | 2 | diverges |
Powers of 2
1 + 2 + 4 + ... + 2ⁿ⁻¹ = 2ⁿ - 1. Ten terms give 1,023. This is why n bits can represent 2ⁿ values and why the legend of rice on a chessboard totals 2⁶⁴ - 1 grains.
How to use it
- Choose the finite sum of n terms or the infinite sum.
- Enter the first term and the common ratio, plus n for the finite case.
- Read the sum, the last term, the partial sums, and how much of the infinite total remains after n terms.
How to read the answer
When |r| < 1 the partial sums climb toward a/(1 - r) and the tail (what is left) shrinks by a factor of r each term. When |r| ≥ 1 the sum keeps growing (or oscillating) without limit, so only finite sums make sense.
Common mistakes and edge cases
- Using the infinite formula when |r| ≥ 1.
- Plugging r = 1 into a(1 - rⁿ)/(1 - r), which divides by zero; the sum is simply n a.
- Forgetting that rⁿ uses n, not n - 1, in the sum formula (n - 1 belongs to the nth term).
Worked examples
2 + 6 + 18 + ... (6 terms)
First term 2, ratio 3
Sum
728
Powers of 2
1 + 2 + ... + 512 = 1,023
Sum
1,023
Halving, finite
1 + 1/2 + ... (10 terms)
Sum
1.99804688
Halving, infinite
Sum to infinity = 2
Sum
2
0.999... = 1
0.9 + 0.09 + ...
Sum
1
Alternating
1 - 1/3 + 1/9 - ... = 0.75
Sum
0.75
Ratio 1
4 + 4 + 4 + 4 + 4 = 20
Sum
20
Divergent infinite
r = 1.5 has no infinite sum
Sum
Error
Frequently asked questions
What is the geometric series formula?+
Sₙ = a(1 - rⁿ)/(1 - r) for the first n terms, and S∞ = a/(1 - r) for the infinite series when |r| < 1.
When does a geometric series converge?+
Exactly when the common ratio satisfies |r| < 1. Then rⁿ shrinks to 0 and the partial sums settle at a/(1 - r).
What is the sum of 1 + 2 + 4 + ... + 2ⁿ⁻¹?+
2ⁿ - 1. For ten terms, 2¹⁰ - 1 = 1,023.
Why does 0.999... equal 1?+
0.999... is the geometric series 0.9 + 0.09 + 0.009 + ... with a = 0.9 and r = 0.1, whose sum is 0.9/(1 - 0.1) = 1.
What happens when r = 1?+
All terms are equal, so the sum is n × a. The usual formula cannot be used because it divides by 1 - r = 0.
Can the ratio be negative?+
Yes. The terms alternate in sign. The infinite sum still exists if |r| < 1: 1 - 1/3 + 1/9 - ... = 3/4.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026