What this calculator does
Summing a series is a different question from listing its terms. An arithmetic or geometric sequence calculator will happily generate the 1st, 2nd or 50th term individually, but adding every one of the first n terms together, the sum of an arithmetic series or the sum of a geometric series, uses its own dedicated formula rather than just totalling terms one at a time.
This calculator covers both cases. An arithmetic series adds a fixed difference at each step, so its terms grow in a straight line and its sum has a simple closed-form formula. A geometric series multiplies by a fixed ratio at each step instead, so its terms grow (or shrink) exponentially, and where that ratio is between −1 and 1, the sum keeps approaching a finite limit even as the number of terms goes to infinity.
The formula
For an arithmetic series, the sum of the first n terms is Sₙ = n/2 × (2a₁ + (n−1)d), where a₁ is the first term and d is the common difference. For a geometric series, Sₙ = a₁(1−rⁿ)/(1−r) for a common ratio r not equal to 1, and where |r| < 1, that same formula converges to a finite sum to infinity, S∞ = a₁/(1−r), as n grows without bound.
| Term | Meaning |
|---|---|
| Common difference (d) | The fixed amount added at each step of an arithmetic series. |
| Common ratio (r) | The fixed multiplier applied at each step of a geometric series. |
| Sₙ | The sum of the first n terms of the series. |
| S∞ | The sum to infinity of a geometric series, which only exists (converges) when the common ratio's absolute value is less than 1. |
The inputs explained
| Field | What to enter |
|---|---|
| Series type | Choose whether the series is arithmetic (adds a fixed amount each term) or geometric (multiplies by a fixed ratio each term). |
| First term (a₁) | The first term of the series. |
| Common difference (arithmetic) or common ratio (geometric) | The common difference if the series is arithmetic, or the common ratio if it is geometric. |
| Number of terms (n) | How many terms to sum, starting from the first term. |
When to use it
Totalling a savings or contribution schedule
A savings plan that adds a fixed extra amount each period is an arithmetic series; summing it gives the total contributed after a set number of periods, without adding each period by hand.
Working out compounding growth over a fixed number of steps
Anything that grows by a constant percentage each step, from repeated doubling to compound interest instalments, is a geometric series, and its sum gives the running total across all those steps combined.
Checking whether an infinite process settles to a finite total
A geometric series with a common ratio between −1 and 1 keeps adding smaller and smaller amounts forever, but the total still converges to a specific finite number, which the sum to infinity shows directly.
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 the sum of an arithmetic series changes with the number of terms
A fixed first term and common difference, across a range of term counts.
| Number of terms | Sum of first n terms (Sₙ) | nth term (aₙ) |
|---|---|---|
| 5 | 55.000 | 17.000 |
| 10 | 185.000 | 32.000 |
| 15 | 390.000 | 47.000 |
| 20 | 670.000 | 62.000 |
| 30 | 1,455.00 | 92.000 |
| 50 | 3,925.00 | 152.000 |
How the sum of a geometric series changes with the common ratio
A fixed first term and term count, across a range of common ratios either side of 1.
| Common ratio | Sum of first n terms (Sₙ) | Sum to infinity |
|---|---|---|
| 0.25 | 6.667 | 6.667 |
| 0.5 | 9.961 | 10.000 |
| 0.75 | 17.998 | 20.000 |
| 1.5 | 246.289 | Diverges: |r| ≥ 1 |
| 2 | 1,275.00 | Diverges: |r| ≥ 1 |
| 3 | 16,400.00 | Diverges: |r| ≥ 1 |
Questions
What is the difference between a sequence and a series?
A sequence is the ordered list of terms itself, such as 5, 8, 11, 14. A series is the sum of those terms added together, such as 5 + 8 + 11 + 14. Generating terms and summing them are related but distinct calculations, which is why this sits alongside, rather than replacing, a dedicated sequence calculator.
When does a geometric series have a sum to infinity?
Only when the common ratio's absolute value is strictly less than 1. In that case each new term is smaller than the last, and the running total approaches a fixed limit rather than growing forever. A ratio of 1 or more, or −1 or less, means the series diverges and has no finite sum to infinity.
What happens with a common ratio of exactly 1 in a geometric series?
Every term is identical to the first, so the sum of n terms is simply n times that first term, which this calculator handles as a special case to avoid dividing by zero in the general formula.
Can the common difference or ratio be negative?
Yes. A negative common difference produces a steadily decreasing arithmetic series, and a negative common ratio produces a geometric series that alternates in sign from term to term, both of which this calculator handles directly.
For the individual terms of an arithmetic series rather than their sum, see the arithmetic sequence calculator. For the individual terms of a geometric series, see the geometric sequence calculator.