1. Introduction
The Fibonacci sequence — 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89... — is defined by the recurrence F(n) = F(n-1) + F(n-2). Introduced in Europe by Leonardo Fibonacci in the 13th century to model the growth of a rabbit population, it is today one of the most studied sequences in mathematics, with deep connections to geometry, biology and number theory.
2. Definition and Properties
Binet's formula gives the n-th Fibonacci term in closed form:
where φ = (1+√5)/2 ≈ 1.618 and ψ = (1-√5)/2 ≈ -0.618. Since |ψ| < 1, the term ψⁿ/√5 tends to 0, and F(n) is the integer nearest to φⁿ/√5.
3. Convergence to φ
The ratio F(n+1)/F(n) converges to φ as n → ∞. This convergence is remarkably fast: by n = 10, the ratio 89/55 ≈ 1.6182 is accurate to 4 decimal places. The convergence is geometric, with a convergence rate of |ψ/φ|² ≈ 0.146 per iteration.
4. Fibonacci in Nature
Fibonacci numbers appear in phyllotaxis (sunflower spirals, pine cone scales), in the number of petals of many flowers (3, 5, 8, 13, 21), in the structure of romanesco broccoli inflorescences, and in the proportions of certain shells. These appearances are linked to the golden angle and to space optimization in continuously growing systems.
5. Generalizations
The Lucas sequence (2, 1, 3, 4, 7, 11, 18...) follows the same recurrence with different initial conditions. Tribonacci sequences (F(n) = F(n-1) + F(n-2) + F(n-3)) and their generalizations converge to other constants. The Padovan sequence and the Perrin sequence are variants with remarkable arithmetic properties.
6. Conclusion
The Fibonacci sequence is a paradigmatic example of how a simple recursive rule generates rich mathematical structure. Its convergence to φ, its appearances in nature and its connections to geometry make it one of the most important sequences in applied mathematics.