Geometric Sequence Calculator (nth Term & Series Sum)
By Santaji GadepatilinUpdated November 4, 2026
A geometric sequence multiplies each term by the same common ratio r — 2, 6, 18, 54 and so on. This geometric sequence calculator returns the nth term and the sum of the first n terms (the geometric series), and solves backwards for the first term, the ratio or the number of terms.
How to use the geometric sequence calculator
Enter the first term, the common ratio and the term number n. The calculator returns the nth term and the sum of the first n terms. Clear a field to solve for it instead — for example, enter the nth term to find the ratio, or a target sum to find how many terms are needed.
Geometric sequence formulas
aₙ = a₁ rⁿ⁻¹ · Sₙ = a₁ (1 − rⁿ) ÷ (1 − r) · S∞ = a₁ ÷ (1 − r) for |r| < 1
Example
First term 2, common ratio 3, n = 6:
- a₆ = 2 × 3⁵ = 2 × 243 = 486
- S₆ = 2 × (1 − 3⁶) ÷ (1 − 3) = 2 × (−728) ÷ (−2) = 728
Terms and partial sums
| n | aₙ (a₁ = 2, r = 3) | Sₙ (a₁ = 2, r = 3) | aₙ (a₁ = 1, r = 0.5) | Sₙ (a₁ = 1, r = 0.5) |
|---|---|---|---|---|
| 1 | 2 | 2 | 1 | 1 |
| 2 | 6 | 8 | 0.5 | 1.5 |
| 3 | 18 | 26 | 0.25 | 1.75 |
| 4 | 54 | 80 | 0.125 | 1.875 |
| 5 | 162 | 242 | 0.0625 | 1.9375 |
| 6 | 486 | 728 | 0.03125 | 1.96875 |
| 7 | 1,458 | 2,186 | 0.015625 | 1.98438 |
| 8 | 4,374 | 6,560 | 0.0078125 | 1.99219 |
| 9 | 13,122 | 19,682 | 0.00390625 | 1.99609 |
| 10 | 39,366 | 59,048 | 0.00195312 | 1.99805 |
The second pair of columns shows a converging series: the partial sums approach S∞ = 1 ÷ (1 − 0.5) = 2.
Notes
- A negative ratio makes the terms alternate in sign.
- Compound interest is a geometric sequence with r = 1 + the interest rate per period — see the compound interest calculator.
Frequently asked questions
What is the formula for the nth term of a geometric sequence?
aₙ = a₁ × rⁿ⁻¹, where a₁ is the first term and r the common ratio.
How do I find the sum of a geometric series?
Sₙ = a₁(1 − rⁿ) ÷ (1 − r) for r ≠ 1. For r = 1 the sum is simply n × a₁.
When does an infinite geometric series converge?
Only when −1 < r < 1. The sum is then S∞ = a₁ ÷ (1 − r) — for example 1 + ½ + ¼ + … = 1 ÷ (1 − 0.5) = 2.
How do I find the common ratio?
Divide any term by the one before it — r = a₂ ÷ a₁. From two terms n − m places apart, r = (aₙ ÷ aₘ)^(1/(n − m)).