Case 1
- Index n (position)
- 10
55
Open with these valuesFibonacci Calculator
55
Result: 55Every term is the sum of the two before it, starting 0, 1, 1, 2, 3, 5, 8, 13. The index counts from zero, so F(0) is 0 and F(10) is 55, not 34. The last term a standard number holds exactly is F(78).
Worked examples
Case 2
- Index n (position)
- 30
832,040
Open with these valuesCase 3
- Index n (position)
- 78
8,944,394,323,791,464
Open with these valuesHow it's calculated
F(n) = F(n − 1) + F(n − 2), F(0) = 0, F(1) = 1
- StepEnter the position n, counting from zero.
- StepThe sequence is built up from 0 and 1.
- StepEach step replaces the pair with the second value and their sum.
- ResultAfter n steps the first value is F(n).
Reference table
| Index n | Reads as | F(n) |
|---|---|---|
| 0 | the first starting value | 0 |
| 1 | the second starting value | 1 |
| 2 | 0 + 1 | 1 |
| 10 | ten steps after the zeroth term | 55 |
| 20 | twice as far, 123 times as large | 6765 |
| 30 | past eight hundred thousand | 832040 |
| 50 | past twelve billion | 12586269025 |
| 78 | the last term held exactly | 8944394323791464 |
Questions
What is the Fibonacci sequence?
It starts with 0 and 1, and every number after that is the sum of the two before it: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34. As a rule, F(n) = F(n − 1) + F(n − 2) with F(0) = 0 and F(1) = 1.
Does this calculator count from 0 or from 1?
From 0: F(0) = 0, F(1) = 1, F(2) = 1, F(3) = 2. The index is the position counting from zero, so entering 10 gives F(10) = 55. Under the other common convention, where 1 is the first term, the same index would give 34.
How is Fibonacci related to the golden ratio?
Divide each term by the one before it and the ratios close in on the golden ratio φ ≈ 1.618. For instance 55 ÷ 34 ≈ 1.6176 and 89 ÷ 55 ≈ 1.6182, and the larger the terms, the closer it sits.
Why does the calculator stop at F(78)?
Beyond that the terms grow larger than a standard computer number can hold exactly. F(78) = 8,944,394,323,791,464 is the last one that fits without rounding error, so the calculator stops there rather than print digits it has lost.
Where does the sequence show up in real life?
In nature — the spiral of a sunflower head, the branching of trees, the arrangement of leaves — and across computer science in algorithms, data structures and search techniques. Its link to the golden ratio makes it a recurring motif in art and design too.
Sources and last check
Information, not professional advice.