What is the Fibonacci Sequence?

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones. It typically starts with 0 and 1, though some definitions start with 1 and 1.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, ...

The Formula

The Fibonacci sequence is defined by the recurrence relation:

F(n) = F(n-1) + F(n-2)

With initial conditions: F(0) = 0, F(1) = 1 (or F(1) = F(2) = 1)

Binet's Formula

The nth Fibonacci number can also be calculated directly using Binet's formula:

F(n) = (φⁿ - ψⁿ) / √5

Where φ (phi) = (1 + √5) / 2 ≈ 1.618 (golden ratio) and ψ = (1 - √5) / 2 ≈ -0.618

Connection to the Golden Ratio

The ratio of consecutive Fibonacci numbers approaches the golden ratio (φ) as the numbers increase:

• 2/1 = 2.000
• 3/2 = 1.500
• 5/3 = 1.667
• 8/5 = 1.600
• 13/8 = 1.625
• 21/13 ≈ 1.615
• 34/21 ≈ 1.619
• ...approaching φ ≈ 1.618...

Fibonacci in Nature

Fibonacci numbers appear remarkably often in nature:

• Flower petals: Lilies have 3, buttercups have 5, delphiniums have 8
• Seed heads: Sunflowers typically have 34 and 55 spirals
• Pinecones: Usually have 8 and 13 spirals
• Pineapples: Often have 8, 13, and 21 spirals
• Nautilus shells: Growth pattern follows the golden spiral

Applications

• Computer Science: Algorithm analysis, data structures
• Finance: Fibonacci retracement in technical analysis
• Art & Design: Aesthetically pleasing proportions
• Biology: Modeling population growth
• Music: Composition structure

First 20 Fibonacci Numbers