Fibonacci Calculator

Find the Nth Fibonacci number, see the sequence, and watch the ratio approach the golden ratio (≈ 1.618033…).

n (term index, F₀ = 0)

Fibonacci formulas

RecurrenceF₀ = 0, F₁ = 1, F_n = F_n−1 + F_n−2

Binet's formulaF_n = φⁿ − ψⁿ√5

ShortcutF_n = round(φⁿ√5)

Golden ratioφ = 1 + √52 ≈ 1.618034, ψ = 1 − √52

Find F₁₀.
1. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
Answer: F₁₀ = 55
Find F₂₀ with the shortcut.
1. φ²⁰ ÷ √5 ≈ 6765.00003
Answer: F₂₀ = 6,765
Ratio of consecutive terms.
1. 89 ÷ 55 ≈ 1.61818
Answer: Approaches φ ≈ 1.618034