Geometric Series Sum Calculator
Sum of a finite or infinite geometric series.
Enter your values
Enter the first term a1 you are working with.
Enter the common 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.
- Finite sum (n terms)
- 3.984375
- Infinite sum
- 4
- Common ratio
- 0.5
Quick answer
A finite geometric series sums to a(1-r^n)/(1-r); an infinite series only converges to a/(1-r) when |r|<1.
S_n = a(1-r^n)/(1-r)
At a glance
| What it does | Sum of a finite or infinite geometric series. |
|---|---|
| Category | Math |
| Inputs needed | First term a1, Common ratio r, Number of terms n |
| Main output | Finite sum (n terms) |
| Formula | S_n = a(1-r^n)/(1-r) |
| Cost | Free — no sign-up, no download |
How to use the Geometric Series Sum Calculator
- 1Enter the first term and the common ratio between terms.
- 2Enter how many terms to sum for the finite sum.
- 3If |r|<1, the infinite sum is also shown since the series converges.
Inputs explained
- First term a1
- Enter the first term a1 you are working with.
- Common ratio r
- Enter the common 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
- First term a12
- Common ratio r0.5
- Number of terms n8
Results
- Finite sum (n terms)3.984375
- Infinite sum4
- Common ratio0.5
Frequently asked questions
When does an infinite geometric series converge?
Only when the absolute value of the common ratio is less than 1.
What if r = 1?
Every term is the same, so the sum is just a times n.
What do I need to enter into the Geometric Series Sum Calculator?
Just 3 values: first term a1, common 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 Geometric Series Sum Calculator work out the answer?
A finite geometric series sums to a(1-r^n)/(1-r); an infinite series only converges to a/(1-r) when |r|<1. It applies the formula S_n = a(1-r^n)/(1-r) and shows the working so you can check each step by hand.
Is the Geometric Series Sum 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.