Find the sum of the first n terms of an arithmetic (constant difference) or geometric (constant ratio) series.
How it works
Arithmetic: S = n/2 × (2a + (n-1)d). Geometric: S = a × (1 – rⁿ) ÷ (1 – r). A series starting at 2 with common difference 3, summed over 10 terms, totals 155.
What this does not include
This covers finite sums of these two classic series types. It doesn’t handle infinite geometric series (which only converge when |r| < 1) or other series types like harmonic series.
How to use this calculator
- Choose arithmetic or geometric.
- Enter the first term, and the common difference or ratio.
- Enter the number of terms.
A worked example
Arithmetic series: first term 2, common difference 3, 10 terms: sum = 155.
Geometric series: first term 1, common ratio 2, 10 terms: sum = 1,023 — geometric series can grow far faster than arithmetic ones over the same number of terms.
What the variables mean
| Variable | Meaning |
|---|---|
| Type | Arithmetic (constant difference) or geometric (constant ratio) |
| a | First term of the series |
| Rate | Common difference (arithmetic) or common ratio (geometric) |
| n | Number of terms to sum |
Edge cases worth knowing
An arithmetic series with the same number of terms grows far more slowly than a geometric one — the two examples above use similar starting values, but the geometric sum ends up more than 6× larger purely from compounding multiplication instead of simple addition.
n must be at least 1 — summing zero terms has no meaningful result, so the calculator declines to show one for n=0.
What’s the difference between arithmetic and geometric series?
An arithmetic series adds the same amount each step; a geometric series multiplies by the same factor each step — the two grow very differently, especially over many terms.
Can the common ratio be negative or a fraction?
Yes — a fractional ratio between -1 and 1 produces a series that shrinks toward zero, while a negative ratio alternates sign each term.
What happens when r equals 1 in a geometric series?
The formula’s denominator would be zero, so the sum is calculated directly as the first term times the number of terms instead — every term is identical when r = 1.
