Geometric Sequence Calculator
Enter the first term, common ratio and n to get the nth term, the partial sum and the infinite sum.
Your numbers
Your results
nth term (aₙ)
nth term (aₙ)
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Sum of n terms (Sₙ)
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Infinite sum
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Geometric Sequence Calculator by TheCalcTools
How the Geometric Sequence Calculator works
Enter the first term, the ratio between consecutive terms and the number of terms.
The nth term is a₁ times r to the power n − 1.
The infinite sum exists only when the absolute ratio is below 1.
The formula
aₙ = a₁ · rⁿ⁻¹Sₙ = a₁(1 − rⁿ) ÷ (1 − r)
Worked example
Using the default values shown in the calculator:
| First term (a₁) | 2 |
|---|---|
| Common ratio (r) | 3 |
| Number of terms (n) | 8 |
- The sequence is 2, 6, 18, 54, …
- a₈ = 2 × 3⁷ = 4,374.
- S₈ = 2 × (1 − 3⁸) ÷ (1 − 3) = 6,560.
Result:
| nth term (aₙ) | 4,374.0000 |
|---|---|
| Sum of n terms (Sₙ) | 6,560.0000 |
| Infinite sum | Diverges (|r| ≥ 1) |
Frequently asked questions
When does an infinite geometric series converge?
When the common ratio is between −1 and 1, because the terms shrink towards zero.
Where do geometric sequences appear?
In compound interest, population growth and radioactive decay.
Sources and further reading
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