Enter a position to get that Fibonacci number and the start of the sequence. Fibonacci numbers begin 1, 1, 2, 3, 5, 8, 13… where each is the sum of the two before it. Because they grow quickly, this tool uses big-integer arithmetic so even the 100th term is exact to the last digit.
Enter a position from 1 upward.
How it works
The sequence is defined by F₁ = 1, F₂ = 1, and Fₙ = Fₙ₋₁ + Fₙ₋₂ for every later term — you add the two previous numbers to get the next. So F₃ = 1 + 1 = 2, F₄ = 1 + 2 = 3, F₅ = 2 + 3 = 5, and on it goes.
The tool builds the sequence term by term with BigInt, which keeps every digit exact rather than rounding once the values pass what a normal number can hold (around the 79th term). Enter a whole position from 1 up to 5,000; the nth number is shown in full, with the first fifteen terms listed so you can see the pattern.
Practical examples
The tenth Fibonacci number
Counting 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, the 10th term is 55. Enter 10 to confirm it and to see the run that leads up to it.
Where ordinary numbers break
The 100th Fibonacci number is 354,224,848,179,261,915,075. A standard calculator rounds the final digits; the exact-integer result here keeps all 21.
The golden ratio appears
Dividing consecutive terms — 55 ÷ 34, then 89 ÷ 55 — closes in on 1.618…, the golden ratio φ. The larger the index, the closer the ratio gets.
Frequently asked questions
What is the Fibonacci sequence?
A list of numbers where each term is the sum of the two before it, starting 1, 1, 2, 3, 5, 8, 13, 21. It is named after Leonardo of Pisa (Fibonacci), who used it to model rabbit populations in 1202.
How is each term calculated?
By the rule Fₙ = Fₙ₋₁ + Fₙ₋₂: add the previous two terms. The first two are both defined as 1, which seeds the whole sequence.
Does the sequence start at 0 or 1?
Both conventions exist. This tool uses the 1-based version F₁ = 1, F₂ = 1. Many maths texts instead start F₀ = 0, F₁ = 1; that just shifts the index by one, so their F₁₀ = 55 matches our position 10 as well.
What is the connection to the golden ratio?
The ratio of consecutive Fibonacci numbers approaches φ ≈ 1.61803… as the terms grow. This is why Fibonacci numbers appear in discussions of the golden ratio and spirals.
How large a term can I compute?
Up to position 5,000. That term has over a thousand digits, and it is computed exactly. Beyond that the output would be unwieldy, so the tool caps the input there.
Why are the results exact and not in scientific notation?
Because the tool uses JavaScript BigInt, which stores whole numbers of any length without rounding. Ordinary floating-point numbers lose precision past about the 79th Fibonacci number.
Where do Fibonacci numbers show up in nature?
They appear in the arrangement of leaves and seeds, the spirals of sunflowers and pinecones, and branching patterns — situations where efficient packing tends to follow these counts.
Is there a formula for the nth term without listing them all?
Yes — Binet’s formula uses the golden ratio: Fₙ = (φⁿ − ψⁿ) ÷ √5. It is elegant but relies on irrational numbers, so for exact large values this tool adds the terms directly instead.
Is my input sent anywhere?
No. The sequence is generated in your browser; the position you enter is never uploaded.
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