What this calculator does
A geometric sequence multiplies by a fixed ratio each step rather than adding a fixed amount. That difference sounds small and produces wildly different behaviour: arithmetic growth is a straight line, geometric growth is an exponential curve.
The ratio decides whether an infinite sum exists at all. With a ratio of 0.5 the terms shrink fast enough that infinitely many of them total just 4.000, while a ratio of 3 makes the sum diverge.
The formula
The nth term multiplies the first term by the ratio raised to n minus 1. The partial sum has a closed form, and when the ratio is between minus one and one the infinite sum converges to the first term over one minus the ratio.
| Term | Meaning |
|---|---|
| Common ratio | The fixed multiplier applied at each step. |
| Convergence | An infinite sum reaching a finite total, which requires the ratio to be below 1 in absolute value. |
| Divergence | The sum growing without limit, which happens at a ratio of 1 or more. |
The inputs explained
| Field | What to enter |
|---|---|
| First term (a₁) | The first term of the sequence. |
| Common ratio (r) | The common ratio. Values between minus one and one give a convergent infinite sum. |
| Term number (n) | Which term to find, and how many terms to sum. |
When to use it
Modelling compound growth
Anything multiplying by a fixed factor each period is geometric.
Summing an infinite series
A shrinking ratio gives a finite total for infinitely many terms.
Understanding exponential behaviour
Comparing against an arithmetic sequence shows how differently the two grow.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How does the ratio change everything?
The same starting term at four common ratios.
| Common ratio | Sum of first n terms (Sₙ) | Sum to infinity |
|---|---|---|
| r = 0.5 | 3.938 | 4.000 |
| r = 1 | 12.000 | Diverges: |r| ≥ 1 |
| r = 2 | 126.000 | Diverges: |r| ≥ 1 |
| r = 3 | 728.000 | Diverges: |r| ≥ 1 |
Questions
Why does a ratio below 1 give a finite infinite sum?
Because the terms shrink geometrically, so each contributes less than the last by a constant factor. The total approaches a limit rather than growing without bound, which is the basis of Zeno's paradox being resolvable.
What happens at a ratio of exactly 1?
Every term equals the first, so the sequence is constant and the sum is simply the first term times n. The general formula divides by one minus the ratio and would fail, so this case is handled separately.
Can the ratio be negative?
Yes, and the terms then alternate in sign. The infinite sum still converges provided the absolute value is below 1, which is why the convergence condition is stated with absolute value.
How does this relate to compound interest?
Directly. A balance growing at a fixed rate is a geometric sequence with the ratio being one plus the rate, which is exactly why compounding produces exponential rather than linear growth.
For the adding equivalent, see the arithmetic sequence calculator. For compounding money, see the compound interest calculator.