🐚 Number Theory & Sacred Geometry
Fibonacci Sequence & Golden Ratio Calculator
Compute the $n^{\text{th}}$ Fibonacci number ($F_n$) using Binet's analytical formula, calculate Golden Section ($\phi \approx 1.6180339887$) proportions, and evaluate sequence ratios.
Sequence Position ($n$)
15th Fibonacci Number (F₁₅)
610
Major Segment ($A$)
61.803
Minor Segment ($B$)
38.197
Lucas Number ($L_n$)
1,364
💡 Binet's Analytic Closed Form:
$$F_n = \frac{\phi^n - \psi^n}{\sqrt{5}} = \frac{(1.618034)^{15} - (-0.618034)^{15}}{2.236068} = 610$$
The Golden Ratio & Binet's Formula
The Fibonacci sequence is defined recursively by $F_0 = 0, F_1 = 1, F_n = F_{n-1} + F_{n-2}$:
- Golden Ratio ($\phi$): $$\phi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887...$$
- Binet's Formula: $$F_n = \frac{\phi^n - (-\phi)^{-n}}{\sqrt{5}}$$
- Limit of Ratios: $$\lim_{n \to \infty} \frac{F_{n+1}}{F_n} = \phi$$