Sum of Series Calculator
Arithmetic, geometric and infinite series sums in one place.
Enter your values
Choose one of: Arithmetic (add d each time), Geometric (multiply by r).
Enter the first term a you are working with.
Enter the common difference d or ratio r you are working with.
Enter the number of terms n you are working with.
Results update instantly as you type — no submit needed. Your values are remembered on this device, and the shareable link reopens the calculator with exactly these numbers.
- Sum of n terms
- 366
- nth term
- 58
- Average term
- 30.5
- Common difference
- 5
- Series type
- Arithmetic
Quick answer
An arithmetic series sums to n/2 × (2a + (n−1)d), and a geometric series to a(1 − rⁿ) ÷ (1 − r), converging to a ÷ (1 − r) when |r| < 1.
Sₙ = n/2(2a + (n−1)d) or a(1 − rⁿ)/(1 − r)
At a glance
| What it does | Arithmetic, geometric and infinite series sums in one place. |
|---|---|
| Category | Math |
| Inputs needed | Series type, First term a, Common difference d or ratio r, Number of terms n |
| Main output | Sum of n terms |
| Formula | Sₙ = n/2(2a + (n−1)d) or a(1 − rⁿ)/(1 − r) |
| Cost | Free — no sign-up, no download |
How to use the Sum of Series Calculator
- 1Pick arithmetic or geometric.
- 2Enter the first term and the difference or ratio.
- 3Enter how many terms you want summed — the infinite sum appears automatically when it converges.
Inputs explained
- Series type
- Choose one of: Arithmetic (add d each time), Geometric (multiply by r).
- First term a
- Enter the first term a you are working with.
- Common difference d or ratio r
- Enter the common difference d or ratio r you are working with.
- Number of terms n
- Enter the number of terms n you are working with.
Worked example
Using the values the calculator loads with:
Inputs
- Series typeArithmetic (add d each time)
- First term a3
- Common difference d or ratio r5
- Number of terms n12
Results
- Sum of n terms366
- nth term58
- Average term30.5
- Common difference5
- Series typeArithmetic
Frequently asked questions
When does an infinite geometric series converge?
Only when the absolute value of the ratio is under 1; then the sum is a ÷ (1 − r).
What if the ratio is exactly 1?
Every term is the same, so the sum is simply a × n.
What do I need to enter into the Sum of Series Calculator?
Just 4 values: series type, first term a, common difference d or ratio r, number of terms n. Every field starts with a realistic example, so you can change one number at a time and watch the result update instantly.
How does the Sum of Series Calculator work out the answer?
An arithmetic series sums to n/2 × (2a + (n−1)d), and a geometric series to a(1 − rⁿ) ÷ (1 − r), converging to a ÷ (1 − r) when |r| < 1. It applies the formula Sₙ = n/2(2a + (n−1)d) or a(1 − rⁿ)/(1 − r) and shows the working so you can check each step by hand.
Is the Sum of Series Calculator free, and do I need an account?
It is completely free with no sign-up, no download and no usage limit. Everything is calculated in your browser, so the numbers you type never leave your device.
Results are estimates for general information. For medical, legal, structural or financial decisions, confirm with a qualified professional.