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The Fibonacci Sequence Calculator computes any term of the Fibonacci sequence instantly using Binet's closed-form formula. Enter a term index $$n$$, and the calculator returns $$F(n)$$, the preceding term $$F(n-1)$$, the next term $$F(n+1)$$, and the ratio $$F(n)/F(n-1)$$ that converges to the golden ratio.
The Fibonacci sequence is one of the most celebrated sequences in all of mathematics. Defined by the recurrence $$F(0) = 0$$, $$F(1) = 1$$, and $$F(n) = F(n-1) + F(n-2)$$ for $$n \geq 2$$, it produces the familiar series 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, and so on. Although named after the Italian mathematician Leonardo of Pisa (Fibonacci), who introduced it to Western Europe in his 1202 book Liber Abaci, the sequence was known centuries earlier in Indian mathematics through the work of Pingala, Virahanka, and Hemachandra in the context of prosody and combinatorics.
The sequence's connection to the golden ratio $$\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887$$ is one of its most remarkable properties. As $$n$$ grows, the ratio of consecutive Fibonacci numbers $$F(n)/F(n-1)$$ converges to $$\varphi$$ with extraordinary speed. By $$n = 12$$, the ratio already agrees with $$\varphi$$ to five decimal places. This convergence is not coincidental — it follows directly from Binet's formula, which expresses the $$n$$-th Fibonacci number as an explicit function of $$\varphi$$.
Fibonacci numbers appear throughout nature and science. The spiral patterns of sunflower seeds, pinecone scales, and nautilus shells follow Fibonacci counts. In computer science, Fibonacci numbers arise in the analysis of algorithms, Fibonacci heaps, and optimal search strategies. In financial mathematics, Fibonacci retracement levels are used in technical analysis. In combinatorics, $$F(n+1)$$ counts the number of ways to tile a $$1 \times n$$ strip with squares and dominoes.
This calculator uses Binet's formula for direct computation, avoiding the slow recursive approach. Binet's formula provides the exact value of any Fibonacci number in constant time, making it practical even for large indices. The calculator handles values up to $$n = 75$$, beyond which floating-point precision limitations may affect results.
The calculator uses Binet's formula, a closed-form expression discovered independently by Jacques Philippe Marie Binet (1843) and Abraham de Moivre (earlier).
Step 1: Define the golden ratio and its conjugate.
$$\varphi = \frac{1 + \sqrt{5}}{2} \approx 1.6180339887$$
$$\psi = \frac{1 - \sqrt{5}}{2} \approx -0.6180339887$$
Note that $$\varphi$$ and $$\psi$$ are the two roots of the characteristic equation $$x^2 = x + 1$$ (equivalently $$x^2 - x - 1 = 0$$).
Step 2: Apply Binet's formula.
$$F(n) = \frac{\varphi^n - \psi^n}{\sqrt{5}}$$
Since $$|\psi| < 1$$, the term $$\psi^n$$ shrinks rapidly toward zero. For $$n \geq 1$$, $$F(n)$$ is the nearest integer to $$\varphi^n / \sqrt{5}$$.
Step 3: Compute adjacent terms. The same formula is applied with $$n-1$$ and $$n+1$$ to obtain $$F(n-1)$$ and $$F(n+1)$$.
Step 4: Compute the golden ratio approximation.
$$\text{ratio} = \frac{F(n)}{F(n-1)}$$
This ratio approaches $$\varphi$$ as $$n$$ increases, illustrating the deep connection between Fibonacci numbers and the golden ratio.
F(n) is the $$n$$-th Fibonacci number. $$F(0) = 0$$, $$F(1) = 1$$, and each subsequent term is the sum of the two preceding terms.
F(n−1) and F(n+1) are the neighboring terms. Together with $$F(n)$$, they demonstrate the defining recurrence: $$F(n+1) = F(n) + F(n-1)$$.
F(n)/F(n−1) shows how quickly the ratio of consecutive terms converges to the golden ratio. For small $$n$$ the ratio oscillates around $$\varphi$$, alternating between values above and below it, but the oscillation dampens rapidly.
Golden Ratio φ is the exact value $$(1 + \sqrt{5})/2$$ displayed for comparison. When the approximation matches $$\varphi$$ to all displayed digits, convergence is effectively complete.
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F(10) = 55 via Binet's formula: (φ¹⁰ − ψ¹⁰)/√5 = (122.9919… − (−0.0081…))/2.2360… ≈ 55. The ratio F(10)/F(9) = 55/34 ≈ 1.6176, already close to φ ≈ 1.6180.
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F(5) = 5, F(4) = 3, F(6) = 8. Verify: 3 + 5 = 8. The ratio 5/3 ≈ 1.6667, which overshoots φ. At this early stage the approximation is still rough but already within 3% of the golden ratio.
Binet's formula is a closed-form expression for the $$n$$-th Fibonacci number: $$F(n) = (\varphi^n - \psi^n)/\sqrt{5}$$, where $$\varphi = (1+\sqrt{5})/2$$ and $$\psi = (1-\sqrt{5})/2$$. It allows direct computation of any Fibonacci number without iterating through all preceding terms.
JavaScript uses 64-bit floating-point numbers (IEEE 754 double precision), which can represent integers exactly up to $$2^{53}$$. The 75th Fibonacci number is approximately $$2.11 \times 10^{15}$$, which is within this range. Beyond $$n \approx 78$$, rounding errors in Binet's formula may produce incorrect integer values.
The convergence is exponential. The error in $$F(n)/F(n-1)$$ relative to $$\varphi$$ decreases by a factor of roughly $$\varphi^{-2} \approx 0.382$$ with each step. By $$n = 15$$, the ratio matches $$\varphi$$ to over 6 decimal places.
By convention, $$F(0) = 0$$ and $$F(1) = 1$$. Some authors use a 1-indexed convention where $$F(1) = 1$$ and $$F(2) = 1$$. This calculator uses the standard 0-indexed convention favored in most modern references.
Yes, extensively. Fibonacci numbers appear in the analysis of the Euclidean algorithm (worst-case inputs are consecutive Fibonacci numbers), Fibonacci heaps (a priority queue data structure), optimal binary search trees, dynamic programming teaching examples, and pseudo-random number generation.
Many natural growth patterns follow Fibonacci numbers because the golden ratio represents the most irrational number — the hardest to approximate by fractions. In phyllotaxis (leaf arrangement), successive leaves or seeds are offset by approximately $$137.5°$$ (the golden angle), which is related to $$\varphi$$. This arrangement maximizes sunlight exposure and packing efficiency.
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