Fibonacci: worked example
| Position n | 10 |
|---|---|
| Fibonacci number F(n) | 55 |
| Ratio F(n) ÷ F(n−1) | 1.6 |
| Sum of the first n terms | 143 |
Fibonacci: formula
Enter the position n of the Fibonacci number you want, counting F(1) and F(2) as 1. Each number is the sum of the two before it: 1, 1, 2, 3, 5, 8 and so on. The calculator also shows the ratio of F(n) to the previous term and the sum of the first n terms, which equals F(n plus 2) minus 1.
Use whole positive positions; very large n produces enormous numbers.
Fibonacci: changing Position n
| Position n | Fibonacci number F(n) | Ratio F(n) ÷ F(n−1) | Sum of the first n terms |
|---|---|---|---|
| 8 | 21 | 1.6 | 54 |
| 9 | 34 | 1.6 | 88 |
| 10 (example) | 55 | 1.6 | 143 |
| 11 | 89 | 1.6 | 232 |
| 12 | 144 | 1.6 | 376 |
Fibonacci: compare input ranges
| Input | Tested from | To | Fibonacci number F(n): low input | Fibonacci number F(n): high input | Swing |
|---|---|---|---|---|---|
| Position n | 9 | 11 | 34 | 89 | 55 (100%) |
Fibonacci: individual input comparisons
| Input | Value | Fibonacci number F(n) | Ratio F(n) ÷ F(n−1) | Sum of the first n terms |
|---|---|---|---|---|
| Position n | 8 | 21 | 1.6 | 54 |
| 9 | 34 | 1.6 | 88 | |
| 10 | 55 | 1.6 | 143 | |
| 11 | 89 | 1.6 | 232 | |
| 12 | 144 | 1.6 | 376 |
Fibonacci: which assumptions move the result?
Fibonacci: values and definitions
| Quantity and definition | Value used |
|---|---|
| Position n — A whole number from 1 to 76. Beyond that the value is too large to hold exactly. | 10 |
| Fibonacci number F(n) | 55 |
| Ratio F(n) ÷ F(n−1) | 1.6 |
| Sum of the first n terms | 143 |
Fibonacci: half and double Position n
| Position n | Fibonacci number F(n) |
|---|---|
| 5 | 5 |
| 10 | 55 |
| 20 | 6,765 |
Common error: Expecting the ratio to hit the golden ratio exactly. It alternates above and below it and only gets closer as the terms grow.
The pattern's appearance in nature is often overstated, and many claimed examples are approximate. Ratios for small n differ visibly from the golden ratio. Some definitions start the sequence at 0.
Large values need many digits, and floating-point rounding appears above about the 70th term.
Number Theory & Precision calculators: Prime Number Calculator, GCF & LCM Calculator, Prime Factorization Calculator, Scientific Notation Calculator, Percent Error Calculator, Number Base Converter Calculator.
Each Fibonacci number is the sum of the two before it, so 1, 1, 2, 3, 5, 8 and on to 55 at the tenth place. The sequence grows by about 1.6 times at each step, and the ratio of neighbors closes in on the golden ratio. The sum of the first ten terms is 143, which is always one less than the term two places further on.
Shells, petals and stock-chart folklore all cite it.
Note that the count starts with F(1) = 1 and F(2) = 1 here. Some texts start with F(0) = 0, which shifts every position by one.
See the golden ratio calculator, the geometric and arithmetic sequence calculators for other patterns, and the factorial page for a faster-growing function. The prime number and modulo pages explore properties of integers. The exponent page shows how fast growth compares with powers.
How is it related to the golden ratio?
The ratio of consecutive Fibonacci numbers approaches the golden ratio, about 1.618, as the numbers get larger.
Where do Fibonacci numbers appear?
In the arrangement of leaves and seeds in some plants, in computer science algorithms and in mathematical puzzles, though some claims about nature are exaggerated.
What is the Fibonacci sequence?
A sequence starting with 1 and 1 in which each number is the sum of the two preceding numbers: 1, 1, 2, 3, 5, 8, 13.