Fibonacci Numbers Calculator

🔢 Fibonacci Calculator

Calculate Fibonacci numbers and explore their properties

Calculation Mode

Nth Number

Sequence

Position (n) Find the nth Fibonacci number (0-1000)

Custom Starting Values (Optional)

F(0) and F(1) - leave default for standard Fibonacci

📊 Results & Properties

Fibonacci calculation results and mathematical properties

Position: F(10)

Fibonacci Number: 55

Ratio to Previous: 1.618

Corresponding Lucas: -

Fibonacci sequence will appear here...

Fibonacci Sequence Guide

The Fibonacci sequence is one of the most famous sequences in mathematics, appearing throughout nature and art. Each number is the sum of the two preceding ones, starting from 0 and 1.

Fibonacci Formula

F(n) = F(n-1) + F(n-2)

where F(0) = 0, F(1) = 1

Binet's Formula: F(n) = (φⁿ - ψⁿ) / √5

where φ = (1+√5)/2 ≈ 1.618 (Golden Ratio)

Golden Ratio: As n increases, the ratio F(n+1)/F(n) approaches the golden ratio φ ≈ 1.618033988749895. This ratio appears throughout art, architecture, and nature.

Lucas Numbers: A related sequence where L(0) = 2, L(1) = 1, and each subsequent number is the sum of the two preceding ones. Lucas numbers share many properties with Fibonacci numbers.

Nature

Spiral patterns in shells, flower petals, pine cones, and tree branches often follow Fibonacci ratios.

Art & Architecture

The golden ratio derived from Fibonacci numbers creates aesthetically pleasing proportions.

Computer Science

Used in algorithms, data structures, and optimization problems.

Financial Markets

Fibonacci retracements are popular in technical analysis for predicting market movements.

Interesting Properties:

Every 3rd Fibonacci number is even
Every 4th Fibonacci number is divisible by 3
Every 5th Fibonacci number is divisible by 5
The sum of the first n Fibonacci numbers equals F(n+2) - 1
F(n)² + F(n+1)² = F(2n+1)

Advanced Fibonacci Numbers Calculator