How the fibonacci calculator works
The Fibonacci sequence starts with 0 and 1, and each later term is the sum of the two before it. It is named after Leonardo of Pisa, who described it in 1202, though Indian mathematicians knew it centuries earlier.
Binet: Fₙ = (φⁿ − ψⁿ) ÷ √5, φ = (1 + √5) ÷ 2
Sum of F₀…Fₙ = Fₙ₊₂ − 1
The ratio of consecutive terms gets closer and closer to the golden ratio φ ≈ 1.6180339887. The calculator works up to 78 terms, the largest that JavaScript can represent exactly with standard numbers.
Worked example
Ten terms give 0, 1, 1, 2, 3, 5, 8, 13, 21, 34. The last term is F₉ = 34, and the sum is 88, which matches F₁₁ − 1 = 89 − 1. The ratio 34 ÷ 21 = 1.619, already within 0.001 of the golden ratio. By 20 terms the ratio is accurate to seven decimal places.
Frequently asked questions
What is the 10th Fibonacci number?
Counting from F₀ = 0, F₁₀ = 55. Counting from a first term of 1, 1, the 10th term is also 55.
What is the golden ratio?
φ = (1 + √5) ÷ 2 ≈ 1.618, the limit of the ratio between consecutive Fibonacci numbers.
Where does the Fibonacci sequence appear in nature?
In spiral counts of sunflower heads, pinecones and pineapples, and in some leaf arrangements.
Why stop at 78 terms?
Beyond that, numbers exceed 2⁵³ and lose exact precision in standard JavaScript arithmetic.
Updated October 2026. Results are estimates for education, not financial or medical advice. All math runs in your browser; nothing you type is sent anywhere.