Overview

The Fibonacci Calculator finds the nth Fibonacci number, generates the sequence, and shows its relationship to the golden ratio. The Fibonacci sequence appears widely in mathematics, nature, algorithm design, and financial technical analysis.

Formula

Fibonacci sequence definition:
F(0) = 0, F(1) = 1
F(n) = F(n-1) + F(n-2) (n ≥ 2)

Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

Binet's formula (closed form):
F(n) = (φⁿ - ψⁿ) / √5
φ = (1 + √5) / 2 ≈ 1.6180 (golden ratio)
ψ = (1 - √5) / 2 ≈ -0.6180

How to Use

1. 1Enter the term number n you want to find.
2. 2The nth Fibonacci number is calculated.
3. 3The entire sequence from F(0) to F(n) is also displayed.
4. 4The ratio between consecutive terms (converging to the golden ratio) is shown.

Tips

• ✔The ratio of consecutive Fibonacci numbers converges to the golden ratio (φ ≈ 1.618).
• ✔Fibonacci numbers appear in sunflower seed arrangements, nautilus shells, and leaf patterns in nature.
• ✔In programming, the Fibonacci sequence is a classic example for recursion and dynamic programming.

FAQ