Patterns that follow a rule
A number sequence is a list of numbers that follows a rule. In an arithmetic sequence you add the same amount each time (2, 5, 8, 11 …). In a geometric sequence you multiply by the same ratio (8, 4, 2, 1 …). A Fibonacci-style sequence makes each term the sum of the two before it.
This calculator finds the n-th term, the sum of the first n terms and a table of the first terms. For a geometric sequence with a ratio between −1 and 1 it also gives the sum to infinity.
How it is calculated
Arithmetic: aₙ = a₁ + (n − 1)d, Sₙ = n ÷ 2 × (2a₁ + (n − 1)d)
Geometric: aₙ = a₁ × rⁿ⁻¹, Sₙ = a₁(1 − rⁿ) ÷ (1 − r)
Sum to infinity = a₁ ÷ (1 − r) when |r| < 1
Worked example
Three sequences
- 2, 5, 8 … 10th term
- 29
- Sum of those 10 terms
- 155
- 8, 4, 2 … sum to infinity
- 16
- Fibonacci, 10th term
- 34
Spotting the type
| Test | Sequence type |
|---|---|
| Differences between terms are equal | Arithmetic |
| Ratios between terms are equal | Geometric |
| Each term is the sum of the previous two | Fibonacci-style |