Fibonacci Sequence and Golden Ratio

Fibonacci Sequence and the Golden Ratio

The Fibonacci sequence starts with 0, 1, and each subsequent number is the sum of the two preceding ones: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

Binet's Formula

You can calculate the nth Fibonacci number directly using Binet's formula:

F(n) = (phi^n - (1-phi)^n) / sqrt(5)

Where phi is the golden ratio (approximately 1.61803).

Examples using the evaluator:

(phi^10 - (1-phi)^10) / sqrt(5) = 55
(phi^20 - (1-phi)^20) / sqrt(5) = 6765

The Golden Ratio (phi)

The golden ratio is built into the evaluator as the constant phi:

phi = (1 + sqrt(5)) / 2 = 1.61803...

Key properties:

phi^2 = phi + 1           (verify: phi^2 - phi)
1/phi = phi - 1           (verify: 1/phi + 1)
phi^2 = 2.618...          (phi^2)

Fibonacci Ratios

As Fibonacci numbers grow, the ratio of consecutive terms approaches phi:

8/5 = 1.6
13/8 = 1.625
21/13 = 1.615...
34/21 = 1.619...
55/34 = 1.617...
89/55 = 1.618...

Applications

• Art and design: The golden rectangle ratio
• Nature: Spiral patterns in sunflowers and shells
• Computer science: Fibonacci heaps, hash tables
• Finance: Fibonacci retracement levels in technical analysis