About this calculator
Choose arithmetic (add the same amount each time) or geometric (multiply by the same amount each time), enter the first term and the common difference or ratio, and get the nth term, the sum of the first n terms, and the first few terms.
For a geometric sequence whose ratio is between −1 and 1, it also gives the sum to infinity.
Worked examples
Real numbers, worked out by the same calculator. Press “Use these numbers” to try one above.
Arithmetic: 2, 5, 8… (10 terms)
- Term 10
- 29
- Sum of the first 10 terms
- 155
- First 10 terms
- 2, 5, 8, 11, 14, 17, 20, 23, 26, 29
In this arithmetic sequence, term 10 is 29 and the first 10 terms add up to 155.
Show the working
- nth term = a + (n − 1) × d = 2 + 9 × 3 = 29.
- Sum = n × (2a + (n − 1)d) ÷ 2 = 10 × (4 + 27) ÷ 2 = 155.
Geometric: 2, 6, 18… (5 terms)
- Term 5
- 162
- Sum of the first 5 terms
- 242
- First 5 terms
- 2, 6, 18, 54, 162
In this geometric sequence, term 5 is 162 and the first 5 terms add up to 242.
Show the working
- nth term = a × rⁿ⁻¹ = 2 × 3^4 = 162.
- Sum = a × (rⁿ − 1) ÷ (r − 1) = 2 × (3^5 − 1) ÷ (3 − 1) = 242.
Geometric: 1, ½, ¼… (20 terms)
- Term 20
- 1.9073e-6
- Sum of the first 20 terms
- 1.999998
- First 10 terms
- 1, 0.5, 0.25, 0.125, 0.0625, 0.03125, 0.015625, 0.007813, 0.003906, 0.001953, …
- Sum to infinity
- 2
In this geometric sequence, term 20 is 1.9073e-6 and the first 20 terms add up to 1.999998.
Show the working
- nth term = a × rⁿ⁻¹ = 1 × 0.5^19 = 1.9073e-6.
- Sum = a × (rⁿ − 1) ÷ (r − 1) = 1 × (0.5^20 − 1) ÷ (0.5 − 1) = 1.999998.
- Because |r| is less than 1, the terms shrink and the sum to infinity is a ÷ (1 − r) = 2.
Arithmetic sequences
Each term is the one before plus a fixed common difference d. The nth term is a + (n − 1)d, and the sum of the first n terms is n × (2a + (n − 1)d) ÷ 2. The sequence 2, 5, 8, 11… has a = 2 and d = 3.
Geometric sequences
Each term is the one before times a fixed common ratio r. The nth term is a × rⁿ⁻¹, and the sum of the first n terms is a(rⁿ − 1) ÷ (r − 1). The sequence 2, 6, 18, 54… has a = 2 and r = 3.
Sum to infinity
If the ratio is between −1 and 1, the terms shrink towards zero and the infinite sum settles to a ÷ (1 − r). For 1, ½, ¼, ⅛… that is 1 ÷ (1 − ½) = 2.
Frequently asked questions
What is the nth term of an arithmetic sequence?
a + (n − 1)d, where a is the first term and d the common difference.
How do I tell if a sequence is arithmetic or geometric?
If the gaps between terms are equal it is arithmetic; if the ratios between terms are equal it is geometric.
When does a geometric series have a sum to infinity?
When the common ratio is between −1 and 1.
What is the sum of 1 to 100?
It is an arithmetic sequence with a = 1, d = 1 and n = 100, so the sum is 5,050.
Formulas tested against hand-worked answers. Last reviewed 29 September 2026. These calculators do arithmetic only; they are not financial, tax or legal advice.