Example 1
- First term: 2
- Ratio: 3
- Terms: 4
80
Terms: 2, 6, 18, 54. Sum: 80.
Geometric series sum calculator
Type the first term, the ratio, and the term count. The geometric series 2, 6, 18, 54 has ratio 3 and sum 80. When the ratio is 1 every term is equal: six fives make 30.
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 times 3. First term 2, ratio 3, four terms, sum 2+6+18+54 = 80. Formula when q ≠ 1: S = a(qⁿ − 1)/(q − 1). 2(81 − 1)/(3 − 1) = 80.
A geometric sequence multiplies. An arithmetic sequence adds a fixed step. The sum 3+6+9+…+21 = 84 lives on the arithmetic series page. Here the step is a product, not plus three.
Fields: First term, Common ratio (multiplier), Number of terms. Type 2, 3, and 4, click Calculate. The term count must be a positive integer. When the ratio is 1 the formula collapses to n × a: 6 × 5 = 30.
A negative ratio flips the sign each time. A ratio 0.5 shrinks the terms. Large n with |q| > 1 grows fast: watch the order of magnitude.
Without any of the three fields, nothing computes. 2, 3, and 4 still give 80 no matter the header.
With a = 1, q = 2, n = 8 the sum is 255 (1+2+4+…+128). With a = 3, q = 3, n = 3 the sum is 3+9+27 = 39.
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
- Type the first term, for example 2.
- Type the common ratio, the multiplier for each next term, for example 3.
- Type the number of terms as a positive integer, for example 4.
- Click Calculate. 2, 3, and 4 give 80. With ratio 1 and six fives the sum is 30.
- A fixed plus-three step lives on the arithmetic series sum. The calculator multiplies by the ratio.
a, q, and n at sum 80
The geometric series 2, 6, 18, 54 has q = 3 and sum 80. At q = 1 and a = 5, n = 6 the sum is 30.
- q
- The ratio. 3 in 2, 6, 18, 54. 2 in 1+2+4+…+128. 1 turns the sum into n × a.
- n
- Term count, a whole number. 4 at sum 80. 8 at 255. 4.5 does not run.
Examples
Example 2
- First term: 5
- Ratio: 1
- Terms: 6
30
Every term is 5. Sum: 6×5 = 30.
Example 3
- First term: 4
- Ratio: −2
- Terms: 5
44
Terms: 4, −8, 16, −32, 64. Sum: 44.
Example 4
- First term: 2
- Common ratio: 3
- Number of terms: 4
80
What is the sum of 2 + 6 + 18 + 54? 80. Ratio 3, four terms.
Related calculators
FAQ
What is the sum of 2, 6, 18, 54?
The sum is 80. Ratio 3, four terms. Formula 2(81 − 1)/2 = 80.
What is the common ratio?
The number you multiply by to get each next term. Here 3: 2 × 3 = 6, 6 × 3 = 18.
What if the ratio is 1, first term 5, n = 6?
The sum is 30. Every term is 5. When q = 1 the formula becomes n × a, so 6 × 5.
How is this different from an arithmetic series?
The calculator multiplies. There you add a fixed step. 3+6+9+…+21 = 84 is on that card.
What is the formula when q ≠ 1?
S = a(qⁿ − 1)/(q − 1). For 2, 3, 4: 2(81 − 1)/2 = 80. At q = 1 the calculator does not divide by zero.
What is 1+2+4+…+128?
The sum is 255. a = 1, q = 2, n = 8. Eight powers of two from 1 to 128.
Can the ratio be a fraction?
Yes. q = 0.5 shrinks the terms. A comma and a period are the same ratio.
Can n be 4.5?
No. The number of terms is a whole number greater than zero. A half term does not enter the formula.
What is 3+9+27?
The sum is 39. a = 3, q = 3, n = 3. Three terms: 3, 9 and 27.
What about a negative ratio?
The signs alternate. The sum still uses the same formula S = a(qⁿ − 1)/(q − 1).
Knowledge sources
The calculator computes the same formula as the definition below.
Page updated in 2026.