Fibonacci Calculator
Calculate values related to the Fibonacci sequence using Binet's formula approximation.
Fib N
55
Formula
## 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.
Exemplo Resolvido
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
Perguntas Frequentes
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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