Fibonacci Numbers
The Fibonacci sequence starts with 0 and 1, and each subsequent term is the sum of the two before it. These numbers appear in nature, art, and many mathematical patterns.
The Sequence
The Fibonacci sequence is defined by F₀ = 0, F₁ = 1, and Fₙ = Fₙ₋₁ + Fₙ₋₂ for n ≥ 2.
The first terms are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, … Each number is the sum of the previous two.
Properties and Patterns
Consecutive Fibonacci ratios approach the golden ratio φ ≈ 1.618. The sum of the first n Fibonacci numbers relates to Fₙ₊₂ − 1.
Every positive integer can be written as a sum of non-consecutive Fibonacci numbers (Zeckendorf representation). Fibonacci numbers also appear in Pascal's triangle diagonals.
Applications
Fibonacci numbers model rabbit population growth (in the classic puzzle), petal counts in flowers, spiral arrangements in shells and pinecones, and algorithm design such as Fibonacci heap structures.
They also connect to the Euclidean algorithm: gcd(Fₙ, Fₙ₋₁) = 1 for all n ≥ 1.
Examples
- F₅ = 5 because 2 + 3 = 5.
- The 10th Fibonacci number is 34.
- The ratio 89/55 ≈ 1.618 approaches the golden ratio.
FAQ
Does the sequence always start with 0?
Some definitions start with 1, 1, 2, 3, … omitting 0. Both conventions are valid; the EveryNum sequence uses F₀ = 0 and F₁ = 1.
Are Fibonacci numbers prime?
Some Fibonacci numbers are prime (2, 3, 5, 13, 89, …), but most are composite. Fibonacci primes are an active area of research.