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Fibonacci Calculator

Step 1 / 3 Nth Fibonacci number
Next

Pick how many terms you want (1 to 200) and this Fibonacci calculator builds the sequence F(1), F(2), F(3) … where each number is the sum of the two before it. It reports the nth Fibonacci number, the running sum of the whole sequence, and the ratio of the last two terms, which converges on the golden ratio. Every value is computed with exact big-integer arithmetic, so even the 200th term, a 42-digit number, is returned without rounding error.

How the calculator works

  1. 1

    Enter the term count

    Type how many Fibonacci numbers you need, from 1 up to 200.

  2. 2

    Choose a mode

    Show the sequence, capped at the first 100 terms, or only the nth term if you care just about the final value.

  3. 3

    Read the results

    See the nth Fibonacci number, the sum of all terms and the golden-ratio approximation, all exact.

The Fibonacci formula

The Fibonacci sequence is defined by a simple recurrence:

F(1) = 1
F(2) = 1
F(n) = F(n-1) + F(n-2)   for n > 2

Each term is the sum of the previous two. This calculator uses the convention F(1) = 1, F(2) = 1, so the sequence begins 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 …

A worked example

To find F(10), add your way up term by term:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55

So F(10) = 55. The sum of those ten terms is 143, and the ratio of the last two, 55 / 34 ≈ 1.6176, is already close to the golden ratio φ ≈ 1.6180339887.

Sequence, sum and golden ratio

n F(n) Sum of F(1)..F(n) F(n) / F(n-1)
5 5 12 1.6667
10 55 143 1.6176
15 610 1596 1.6181
20 6765 17710 1.6180

A neat identity: the sum of the first n Fibonacci numbers equals F(n+2) − 1. For n = 10 that is F(12) − 1 = 144 − 1 = 143, matching the table.

Common pitfalls

  • Indexing differs. Some textbooks start at F(0) = 0. This tool uses F(1) = 1, F(2) = 1, so shift the index by one if you are comparing against an F(0)-based table.
  • Floating-point overflow. Computing large Fibonacci numbers with ordinary floats loses precision past about F(78). This calculator uses exact big-integer math, so F(200) (a 42-digit number) is returned in full.
  • The golden ratio is a limit, not an equality. F(n) / F(n-1) only approaches φ; it is never exactly equal for finite n. Small n values are noticeably off.

Frequently Asked Questions

A series of numbers where each term is the sum of the two before it, starting 1, 1, 2, 3, 5, 8, 13, 21 … It appears throughout mathematics and nature, from leaf arrangements to spiral shells.

By the recurrence F(n) = F(n-1) + F(n-2). This tool adds terms iteratively with exact big-integer arithmetic, so even F(200) is computed without rounding.

As n grows, F(n) / F(n-1) converges to the golden ratio φ ≈ 1.6180339887, a fundamental property of the sequence proven from its closed-form (Binet) expression.

It is not stored or shared anywhere. The number you enter is sent to our servers to compute the results, and in the step-by-step flow it also travels in the page link between steps. It is used for nothing else.

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