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Math Tools

Fibonacci Sequence Generator

Generate, explore, and analyze Fibonacci numbers

Sequence Generator

Generate the Fibonacci sequence with custom parameters

Find Nth Fibonacci Number

Calculate any specific term in the Fibonacci sequence

F₀ = 0, F₁ = 1, F₂ = 1, F₃ = 2, ...

Quick Lookup

Fibonacci Number Checker

Check if a number is in the Fibonacci sequence

Enter any positive integer

Check Multiple Numbers

Golden Ratio Explorer

Explore the connection between Fibonacci and the Golden Ratio

φ
1.6180339887...
The Golden Ratio (Phi)
φ (Phi)
1.618033989
(1 + √5) / 2
1/φ (Reciprocal)
0.618033989
φ - 1
φ² (Squared)
2.618033989
φ + 1

Fibonacci Ratio Convergence

As n increases, Fₙ₊₁/Fₙ approaches the Golden Ratio

Binet's Formula
Fn = (φⁿ - ψⁿ) / √5

where ψ = (1 - √5) / 2 ≈ -0.618

Golden Ratio Calculator
Longer Section (a)
61.80
Shorter Section (b)
38.20
Ratio (a/b)
1.618

Fibonacci Reference

Properties, formulas, and applications of Fibonacci numbers

First 30 Fibonacci Numbers

Key Properties
Definition
Fn = Fn-1 + Fn-2
Sum of First n Terms
∑Fk = Fn+2 - 1
Sum of Squares
F1² + F2² + ... + Fn² = Fn × Fn+1
GCD Property
gcd(Fm, Fn) = Fgcd(m,n)
Real-World Applications
Nature
  • Flower petals (lilies: 3, buttercups: 5)
  • Seed spirals in sunflowers
  • Pinecone scales
  • Leaf arrangements
Art & Design
  • Golden rectangle proportions
  • Photography composition
  • Architecture (Parthenon)
  • Logo design
Computer Science
  • Fibonacci heaps
  • Fibonacci search
  • Pseudorandom generators
  • Algorithm analysis

Understanding Fibonacci Numbers

What Are Fibonacci Numbers?

A sequence where each number is the sum of the two preceding ones.

  • Named after Leonardo of Pisa
  • Starts: 0, 1, 1, 2, 3, 5, 8, 13...
  • Found throughout nature
  • Connected to the Golden Ratio
The Golden Connection

Consecutive Fibonacci ratios approach the Golden Ratio.

  • φ ≈ 1.6180339887...
  • F(n+1)/F(n) → φ as n → ∞
  • φ² = φ + 1
  • 1/φ = φ - 1
Growth Rate

Fibonacci numbers grow exponentially.

  • F(n) ≈ φⁿ / √5
  • Doubles roughly every 1.44 terms
  • F(100) has 21 digits
  • F(1000) has 209 digits