What this calculator does
Each Fibonacci number is the sum of the two before it, starting from 0 and 1. That trivial rule produces a sequence that appears throughout mathematics and, genuinely if less often than claimed, in nature.
The ratio of consecutive terms converges on the golden ratio remarkably fast. By the tenth term it already reads 1.618 to three decimal places, and it continues to tighten from there.
The formula
The sequence is built up term by term from the two starting values. The running sum uses the identity that the sum to F(n) equals F(n+2) minus 1.
| Term | Meaning |
|---|---|
| Fibonacci number | A term in the sequence where each equals the sum of the two before it. |
| Golden ratio | The limit of the ratio of consecutive terms, approximately 1.618. |
| Sum identity | The sum of the first n terms equals F(n+2) less 1. |
The inputs explained
| Field | What to enter |
|---|---|
| Term index n (F(0), F(1), …) | Which term to compute, counting from F(0) = 0. |
When to use it
Finding a specific term
The sequence grows fast and terms become awkward to compute by hand.
Seeing the golden ratio emerge
The ratio of consecutive terms converges visibly within ten steps.
Checking the sum identity
The running sum relates to a later term in the sequence.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How fast does the ratio converge?
Four terms and the ratio at each.
Questions
Why does the ratio converge to the golden ratio?
Because the defining recurrence, divided through by the previous term, approaches an equation whose positive solution is exactly phi. The convergence is geometric, which is why so few terms are needed.
At what point does the Fibonacci sequence pass 4 million?
At the 34th term. F(33) is 3,524,578, still short of it, and F(34) is 5,702,887, the first Fibonacci number above 4,000,000. The step between the two is that large because the sequence grows geometrically, multiplying by roughly the golden ratio each time.
Does Fibonacci really appear in nature?
In some places genuinely. Sunflower seed spirals and pine cone scales do follow Fibonacci counts, because that arrangement packs seeds most efficiently. Many other claimed appearances, particularly in art and architecture, do not survive measurement.
Is there a direct formula?
Yes, Binet's formula, which expresses F(n) using powers of the golden ratio. Remarkably it produces exact integers despite involving irrational numbers throughout, because the irrational parts cancel.
What is the sum identity?
The sum of F(0) through F(n) equals F(n+2) minus 1. Summing the first ten terms gives 143, which is F(12) of 144 less 1, and the pattern holds for every n.
For the ratio it converges to, see the golden ratio calculator. For sequences that multiply instead, see the geometric sequence calculator.