A series is what you get by adding the terms of a sequence. Closed-form sums save adding hundreds of terms by hand and appear in loan formulas, algorithm analysis (1 + 2 + … + n steps) and physics.
How to use the Sum of Series Calculator
- Series
- First term a
- Common difference d
- Common ratio r
- Number of terms n
Results update as you type, and everything is calculated in your browser.
How it works
Choose the series and enter its parameters. For geometric series with |r| < 1 the infinite sum is also shown.
Inputs explained
| Field | What to enter |
|---|---|
| Series | Choose one: Arithmetic, Geometric, 1 + 2 + … + n, 1² + 2² + … + n² … |
| First term a | A number |
| Common difference d | A number |
| Common ratio r | A number |
| Number of terms n | A number |
Example values
How the main result changes when First term a changes and everything else stays at the example values above:
| First term a | Sum Sₙ |
|---|---|
| 0.5 | 95 |
| 0.75 | 97.5 |
| 1 | 100 |
| 1.25 | 102.5 |
| 1.5 | 105 |
| 2 | 110 |
Formula
Arithmetic Sₙ = n(2a + (n − 1)d)/2 · Geometric Sₙ = a(1 − rⁿ)/(1 − r)
Σk = n(n + 1)/2 · Σk² = n(n + 1)(2n + 1)/6 · Σk³ = (n(n + 1)/2)²
Worked example
Inputs — Series: Arithmetic; First term a: 1; Common difference d: 2; Number of terms n: 10.
Results — Sum Sₙ = 100; Last term = 19.
Key terms
- Series
- The sum of the terms of a sequence.
Frequently asked questions
What is the sum of 1 to 100?
5,050.
What is the sum of the first n odd numbers?
n².
Why is Σk³ = (Σk)²?
A beautiful identity, provable by induction or with a picture of nested squares.
When does a geometric series have an infinite sum?
When |r| < 1.