Fibonacci Sequence
Generate the Fibonacci sequence where each number is the sum of the two before it.
Found throughout nature and mathematics.
The Formula
F(n) = F(n-1) + F(n-2) where F(0) = 0, F(1) = 1
Each Fibonacci number is the sum of the two numbers before it. The sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...
Binet's closed-form formula: F(n) = (φⁿ - ψⁿ) / √5, where φ = (1+√5)/2 ≈ 1.618 (the golden ratio) and ψ = (1-√5)/2 ≈ -0.618.
Variables
| Symbol | Meaning |
|---|---|
| F(n) | The nth Fibonacci number |
| n | Position in the sequence (starting from 0) |
| φ | Golden ratio (approximately 1.61803) |
Example 1
Find the 10th Fibonacci number
F(0)=0, F(1)=1, F(2)=1, F(3)=2, F(4)=3, F(5)=5
F(6)=8, F(7)=13, F(8)=21, F(9)=34, F(10)=55
F(10) = 55
Example 2
Use Binet's formula to find F(12)
F(12) = (1.618¹² - (-0.618)¹²) / √5
F(12) = (321.997 - 0.00319) / 2.236
F(12) = 144
When to Use It
Use the Fibonacci sequence when:
- Studying patterns in nature (sunflower seeds, pinecones, shells)
- Analyzing algorithm complexity (recursive algorithms)
- Working with the golden ratio in art and architecture
- Applying Fibonacci levels in financial trading