Key takeaways
- Sₙ = a(rⁿ - 1) / (r - 1) totals the first n terms of a geometric sequence.
- The exponent here is n, not (n - 1) — a different exponent than the single-term formula.
- If r = 1, every term is identical, so the sum is simply a × n.
- A common ratio greater than 1 makes the total grow explosively as n increases.
In this guide
What is a geometric series?
It's the sum you get from adding every term of a geometric sequence together, from the first term up to the nth term. Because each term is a multiple of the last, the total can grow very quickly for ratios greater than 1.
Why does the sum formula matter?
Multiplying term by term and adding as you go is slow, and it's easy to lose track partway through. This shortcut totals an entire geometric series in one calculation, no matter how many rounds or terms are involved.
Six rounds of doubling starting from 3 don't just add up round by round — the running total across all six rounds is far bigger than any single round's number, because every earlier round is still counted too.
The formula
| Symbol | What it means | Example |
|---|---|---|
| Sₙ | The total of the first n terms | 189 |
| a | The first term | 3 |
| r | The common ratio between terms (not equal to 1) | 2 |
| n | How many terms you're adding up | 6 |
How to find the sum step by step
- 1Identify a, r, and n. The first term, common ratio, and number of terms.
- 2Raise r to the power n.
rⁿ. - 3Subtract 1 from rⁿ, and separately subtract 1 from r. This gives
(rⁿ - 1)and(r - 1). - 4Multiply a by (rⁿ - 1), then divide by (r - 1).
Enter your own a, r, and n below to see the total worked out step by step.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Dividing by zero. If r = 1, every term is identical, so the sum is simply
a × ninstead — the main formula doesn't apply. - Confusing this with the arithmetic series sum, which adds a fixed amount each term instead of multiplying — see the arithmetic series calculator for that case.
- Forgetting to divide by (r - 1) after multiplying, which leaves the answer far too large.
Tips and tricks
- For a single term instead of a running total, use the geometric sequence nth term calculator.
- This formula also totals the combined value of repeated interest payments — pair it with the compound interest calculator.
- Check small cases by hand: n = 1 should always give Sₙ = a.
Where you'll use it in real life
- Social media: the total cumulative reach of a message that keeps doubling round after round.
- Finance: the combined value of a series of compounding payments over time.
- Biology: total population across several doubling generations.
- Physics: the total distance a bouncing ball travels across many shrinking bounces.
Quick summary
The geometric series formula adds up every term from the first to the nth without listing them all: raise r to the power n, subtract 1, multiply by a, then divide by (r - 1). Try the calculator above with your own numbers to see the full working.
Worked example
Continuing the video-sharing example: 3 people start it, and the number of new viewers doubles every round for 6 rounds. How many people have seen it in total across all 6 rounds?
- 01Sₙ = a(rⁿ - 1) / (r - 1)
- 02S6 = 3(2^6 - 1) / (2 - 1)
- 03S6 = 3(63) / 1
- 04S6 = 189
These numbers are pre-filled in the calculator above.
Frequently asked questions
How do I find the sum of a geometric series?
Use Sₙ = a(rⁿ - 1) / (r - 1), where a is the first term, r is the common ratio, and n is the number of terms.
What if the common ratio is 1?
The formula would divide by zero, so it doesn't apply. Instead, every term equals a, so the sum is just a × n.
Can the sum of a geometric series be negative?
Yes, if the first term is negative, or if the ratio is negative and the terms alternate sign in a way that leaves the total negative.
What's the difference between the nth term formula and the sum formula?
The nth term formula gives the value of one specific term. The sum formula adds up every term from the first to the nth.
