Number sequence calculator
The nth term and sum of arithmetic, geometric and Fibonacci sequences, or the next numbers in a list.
About this calculator
This number sequence calculator finds any term of an arithmetic, geometric or Fibonacci-type sequence, and the sum of the first n terms.
Enter a list of numbers instead, and it finds the pattern, gives the rule and works out the next five numbers.
How it works
An arithmetic sequence adds the same difference d each time: aₙ = a₁ + (n − 1)d. A geometric one multiplies by the same ratio r: aₙ = a₁ × rⁿ⁻¹. In a Fibonacci-type sequence each term is the sum of the two before it.
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Worked example
- Sequence: Arithmetic: add the same amount
- First term: 3
- Common difference: 4
- Term number (n): 10
Value of term n: 39
Common questions
How do I find the nth term of an arithmetic sequence?
Use aₙ = a₁ + (n − 1)d. For 3, 7, 11 and so on, the 10th term is 3 + 9 × 4 = 39.
What is the sum of a geometric series?
Sₙ = a₁(1 − rⁿ) ÷ (1 − r). If the ratio is between −1 and 1, the sum to infinity is a₁ ÷ (1 − r).
What are the first Fibonacci numbers?
0, 1, 1, 2, 3, 5, 8, 13, 21, 34 and 55. Each one is the sum of the two before it.
Sources and further reading
The story behind it
Where the idea came from: Leonardo of Pisa, known as Fibonacci, described his famous sequence in 1202 in his book Liber Abaci, through a puzzle about breeding rabbits.
The ratio of neighbouring Fibonacci numbers gets closer and closer to the golden ratio, about 1.618. The same numbers turn up in the spirals of sunflower heads and pine cones.
Why you can trust this calculator
- It runs in your browser. Nothing you type into it is sent anywhere.
- The method is explained on this page, so you can check it yourself.
- Free to use, with no sign-up.
- Results are estimates, so please double-check anything important.
- Spotted a mistake? Tell us and we'll look into it.
After 15 years as a mathematics educator, I built CalculStore to give students, teachers, professionals and anyone working with numbers simple, reliable tools.
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