The Mathematics of the Fibonacci Sequence

Fibonacci sequence: 0,1,1,2,3,5,8,13,21,34,55,89,144,233,377,… Each term is sum of previous two. The ratio Fₙ₊₁/Fₙ approaches φ≈1.618. The sequence appears in pine cones, sunflowers, and rabbit populations. It also satisfies Fₙ = (φⁿ – ψⁿ)/√5, where ψ = (1‑√5)/2. Fibonacci numbers are used in algorithms (Fibonacci search, Fibonacci heaps) and in the analysis of Euclid’s algorithm. The sequence is a simple linear recurrence that generates endless properties.