Fibonacci Calculator

Getting an answer takes four quick steps. The tool needs just one input, the term position.

1. Type the term position n in the "Term position (n)" field. Use 10 for the 10th term, or 50 for the 50th. The field accepts whole numbers from 0 to 10000.
2. Press Calculate, or press Enter. The result appears with the sequence, the term sum, and the ratio to the previous term.
3. Read the step-by-step breakdown under the answer. It shows each addition the recurrence performs.
4. Press Reset to clear the form and try another position.

The Fibonacci sequence is a list of whole numbers where each number is the sum of the two numbers before it. It starts 0, 1, and then every new term adds its two neighbors, so the rule never changes no matter how far you go.

This page uses the standard convention F(0) = 0 and F(1) = 1. State this convention every time you compare answers across sources, because some books start at F(1) = 1 and F(2) = 1, which shifts every index by one.

The first 20 terms are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, and 4181. Each term after the first few is roughly 1.6 times the one before it, which is why the list grows so fast.

Two formulas produce the same terms. The recurrence builds them step by step, and Binet's formula jumps straight to any term.

The Recurrence Relation

The recurrence is F(n) = F(n-1) + F(n-2), with F(0) = 0 and F(1) = 1. Here F(n) is the term you want, and F(n-1) and F(n-2) are the two terms right before it. To find F(10), build up from the start. F(2) = 1 + 0 = 1, F(3) = 1 + 1 = 2, F(4) = 2 + 1 = 3, F(5) = 3 + 2 = 5, F(6) = 5 + 3 = 8, F(7) = 8 + 5 = 13, F(8) = 13 + 8 = 21, F(9) = 21 + 13 = 34, and F(10) = 34 + 21 = 55. The calculator on this page follows exactly this chain.

Binet's Formula

Binet's formula computes F(n) directly: F(n) = (phi^n - psi^n) / sqrt(5), where phi = (1 + sqrt(5)) / 2, about 1.6180339887, and psi = (1 - sqrt(5)) / 2, about -0.6180339887. Since psi^n shrinks toward zero as n grows, a handy shortcut is to round phi^n / sqrt(5) to the nearest whole number. For n = 10, phi^10 / sqrt(5) is about 54.999, which rounds to 55. Carry full precision through the powers and round only at the very end, or large terms drift. The formula divides by the square root of 5. To explore that step on its own, try the square root calculator.

The Sum Rule

The terms add up neatly: F(0) + F(1) + ... + F(n) = F(n+2) - 1. Add the first 10 terms and you get 0 + 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 = 88. Since F(11) = 89, the rule predicts 89 - 1 = 88, which matches. The calculator shows this sum next to every answer, so you can check any long list in one glance.

Divide each term by the one before it and watch what happens. The ratios run 1, 2, 1.5, 1.667, 1.6, and 1.625, swinging above and below a fixed value and landing closer each time. That value is the golden ratio, phi, about 1.6180339887.

Comparing consecutive terms is a ratio comparison like any other, and our ratio calculator handles the same kind of side-by-side comparison for everyday numbers. F(50) / F(49) reads 1.61803399, correct to eight decimal places.

The ratio never reaches phi exactly for a finite term, but it gets closer without limit. That slow, alternating approach is why phi is sometimes called the most irrational number.

Sunflower seed heads often show 34 spirals one way and 55 the other, and cactus stems carry 13 spirals. New seeds pack at the golden angle, about 137.5 degrees, because that spacing leaves the fewest gaps.

The same numbers turn up in flower petals, pinecone scales, and tree branching. Artists and architects have borrowed the proportion for pleasing rectangles, and programmers meet the sequence in recursion, the classic example of a naive recursive function repeating work while a loop runs in linear time.

Leonardo of Pisa, known as Fibonacci, introduced the sequence to Europe in his 1202 book Liber Abaci, through the rabbit-breeding problem that asks how many pairs of rabbits appear in a year when each pair breeds a new pair monthly. Indian scholars had described the same numbers centuries earlier while studying the rhythms of poetry.

The most common error is the indexing shift. If one source counts F(1) = 1, F(2) = 1 and another counts F(0) = 0, F(1) = 1, the "10th term" differs by one position, so always confirm the convention first.

Rounding Binet's formula too early is the next trap. Truncating phi to a few decimals compounds into a wrong integer for large n, so carry full precision and round once at the end.

Also, do not expect the ratios to settle instantly. Early ratios like 1 or 2 sit far from phi, and the convergence only tightens past n = 15 or so. Finally, do not confuse Fibonacci numbers with Lucas numbers, which follow the same recurrence from different seeds, L(0) = 2 and L(1) = 1, producing 2, 1, 3, 4, 7, 11, and so on.

If you came here for trading, you want a different tool. A Fibonacci retracement calculator plots price levels from a swing high and low at 23.6%, 38.2%, 50%, 61.8%, and 78.6%, with extensions at 127.2%, 161.8%, and 261.8%. Those levels borrow ratios from the sequence, but this page computes the sequence itself, the nth term, the running list, and the sum.