Geometric sequences multiply by the same ratio each step, like 2, 6, 18, 54, …
How to use the Geometric Sequence Calculator
- First term (a₁)
- Common ratio (r)
- Term number (n)
Results update as you type, and every field accepts the units shown in its menu.
How it works
Raise the ratio to n − 1 and multiply by the first term. The sum uses the closed formula for a geometric series.
Inputs explained
| Field | What to enter |
|---|---|
| First term (a₁) | A number |
| Common ratio (r) | A number |
| Term number (n) | A number |
Example values
How the main result changes when First term changes and everything else stays at the example values above:
| First term (a₁) | nth term |
|---|---|
| 1 | 2,187 |
| 1.5 | 3,280.5 |
| 2 | 4,374 |
| 2.5 | 5,467.5 |
| 3 | 6,561 |
| 4 | 8,748 |
Formula
aₙ = a₁ × r^(n − 1)Sₙ = a₁(1 − rⁿ) ÷ (1 − r)S∞ = a₁ ÷ (1 − r), |r| < 1Worked example
Inputs — First term (a₁): 2; Common ratio (r): 3; Term number (n): 8.
Results — nth term = 4,374; Sum of first n terms = 6,560.
Key terms
- Common ratio
- The fixed multiplier between consecutive terms.
Frequently asked questions
When does the infinite sum exist?
Only when the ratio is between −1 and 1.
Where are geometric sequences used?
Compound interest, population growth and fractals.
What formula does the Geometric Sequence Calculator use?
aₙ = a₁ × r^(n − 1).