Fibonacci Calculator Formula
Understand the math behind the fibonacci calculator. Each variable explained with a worked example.
Formulas Used
Fib N
fib_n = round(pow((1 + sqrt(5)) / 2, n) / sqrt(5))Golden Ratio
golden_ratio = (1 + sqrt(5)) / 2Fib N Minus 1
fib_n_minus_1 = n >= 1 ? round(pow((1 + sqrt(5)) / 2, n - 1) / sqrt(5)) : 0Ratio
ratio = n >= 2 ? round(pow((1 + sqrt(5)) / 2, n) / sqrt(5)) / round(pow((1 + sqrt(5)) / 2, n - 1) / sqrt(5)) : 0Variables
| Variable | Description | Default |
|---|---|---|
n | Position (n) | 10 |
How It Works
Fibonacci Sequence
Definition
F(0) = 0, F(1) = 1, F(n) = F(n-1) + F(n-2)
Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
Binet's Formula
F(n) = (phi^n - psi^n) / sqrt(5)
where phi = (1 + sqrt(5))/2 (golden ratio) and psi = (1 - sqrt(5))/2.
For large n, psi^n approaches 0, so F(n) ≈ phi^n / sqrt(5).
Golden Ratio
As n grows, F(n)/F(n-1) approaches the golden ratio phi ≈ 1.6180339887.
Worked Example
Find the 10th Fibonacci number.
- 01F(10) = phi^10 / √5
- 02= 1.61803^10 / 2.23607
- 03≈ 122.992 / 2.236
- 04≈ 55
- 05Sequence up to F(10): 0,1,1,2,3,5,8,13,21,34,55
Frequently Asked Questions
What is the Fibonacci sequence?
The Fibonacci sequence starts with 0 and 1, and each subsequent number is the sum of the two preceding ones: 0, 1, 1, 2, 3, 5, 8, 13, 21, ...
What is the golden ratio?
The golden ratio (phi ≈ 1.618) is the limit of the ratio of consecutive Fibonacci numbers. It appears in art, architecture, and nature.
Where do Fibonacci numbers appear in nature?
Fibonacci numbers appear in flower petals, seed spirals in sunflowers, pinecones, pineapples, and branching patterns in trees.
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