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Fibonacci Number Calculator

Find any Fibonacci term.

Result

6,765

How it is calculated

The tool builds the sequence step by step with Fn = Fn−1 + Fn−2, starting from F1 = 1 and F2 = 1, and shows term number n. So the first terms are 1, 1, 2, 3, 5, 8, 13, 21 and the 10th term is 55. Numbering starts at 1: the sequence's 0 is not available as an input.

n must be a whole number from 1 to 92; decimals, 0, negative numbers and anything above 92 give no result. The arithmetic uses exact integers, so every digit is correct up to the 92nd term, 7,540,113,804,746,346,429, which is the largest term that fits in a 64-bit integer. The result is displayed with thousands separators.

Worked example

Ask for the 20th term (n = 20). The sequence runs 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181. The 18th term is 2584 and the 19th is 4181, so the 20th is 2584 + 4181 = 6765. The tool shows 6,765. As a check, 6,765 ÷ 4,181 = 1.618, close to the golden ratio.

Try this example →

Common mistakes

  • Expecting n = 0 to work or n = 1 to give 0. Here the 1st and 2nd terms are both 1 and n = 0 is rejected, so the tool never returns the sequence's leading 0.
  • Entering a decimal or a value above 92. Only whole positions from 1 to 92 work; anything else returns nothing, so a term like F(100) must be computed with big-integer software.
  • Off-by-one errors in indexing. In a list starting 1, 1, 2, 3, 5, the 5th term is 5 and the 6th is 8.
  • Confusing the Fibonacci number with the sum of the sequence up to n. The tool gives the single term, not the running total.

About this calculator

The Fibonacci sequence — 1, 1, 2, 3, 5, 8, 13, 21, and onward, where each term is the sum of the two before it (F_n = F_n−1 + F_n−2) — was introduced to Europe by Leonardo of Pisa, known as Fibonacci, in his 1202 book Liber Abaci, though the pattern had already been described earlier in Indian mathematics. As the sequence grows, the ratio between consecutive terms converges toward the golden ratio, φ ≈ 1.618, which is why Fibonacci numbers turn up in discussions of the golden ratio's appearance in art, architecture, and nature.

The sequence is a staple of computer science education as a classic example for teaching recursion and dynamic programming, and it shows up in biology in phyllotaxis — the spiral arrangement of leaves, seeds, and pinecone scales often follows consecutive Fibonacci numbers because that spacing pattern packs growth most efficiently.

The position n can be any whole number from 1 to 92, counting from F(1) = F(2) = 1.

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Comments (1)

  • Editorial team

    Divide a term by the one before it and the ratio closes in on 1.618, the golden ratio. You can see it in the tool: the 21st term is 10,946, the 20th is 6,765, and 10,946 divided by 6,765 is 1.6180. Sources that start the sequence at 0 give the same value for any term from the first upward. Do you need terms beyond the 92nd, and what for?

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