Example 1 — growing sequence
- First term: 2
- Ratio: 3
- Terms: 4
80
Terms: 2, 6, 18, 54. Sum: 80.
Geometric series sum
Enter the first term, the constant multiplier, and how many terms to add. Get the sum without a long hand list.
A geometric sequence multiplies by the same number (the common ratio) from term to term. Here we sum the first n terms. If the ratio is 1, every term equals the first — the sum is simply n times the first term.
Result
Fill in the fields to see the sum.
What it means
Example: 2, 6, 18, 54… Each next term is the previous one times 3. First term 2, ratio 3.
The sum adds those terms. For many terms a formula is easier than multiplying everything by hand.
A negative ratio flips the sign each step. A large ratio makes the sum grow quickly — be careful with large n.
Where it helps
- School work on geometric sequences.
- Simple models of repeated percentage growth.
- Checking a sum without writing a long list.
- Comparing with arithmetic sequences: multiply here, add there.
How to use it
- Enter the first term.
- Enter the common ratio (the multiplier).
- Enter how many terms (a whole number greater than zero).
- Read the sum. If the ratio is 1, the answer is simply n × first term.
Examples
Example 2 — ratio 1
- First term: 5
- Ratio: 1
- Terms: 6
30
Every term is 5. Sum: 6×5 = 30.
Example 3 — negative ratio
- First term: 4
- Ratio: −2
- Terms: 5
44
Terms: 4, −8, 16, −32, 64. Sum: 44.
FAQ
What is a geometric sequence?
A list where each next term is the previous one times a fixed number. Example: 3, 6, 12, 24 (ratio 2).
What is the common ratio?
The fixed multiplier between neighbors. It can be a fraction or negative.
How is this different from an arithmetic sequence?
Here you multiply. In an arithmetic sequence you add a fixed difference.
Why must n be a whole number?
It counts how many terms are on the list — you cannot have “half a term”.
What if the ratio is 1?
Every term equals the first. Sum = n × first term.
What if the ratio is 0?
The first term stays, later terms are zero (if n > 1). The sum equals the first term.
Why does a large n explode?
Repeated multiplication grows very fast. That is normal for geometric sequences.
Are negative ratios allowed?
Yes. The signs of the terms will alternate.