Fibonacci Sequence Calculator
The Fibonacci sequence starts 0, 1 and every term after is the sum of the two before: 0, 1, 1, 2, 3, 5, 8, 13, 21…. Enter any index n between 0 and 1500 and this tool returns F(n), F(n−1), the consecutive-term ratio F(n) ÷ F(n−1) — which converges to the golden ratio φ ≈ 1.618 — and the first 30 terms, all using BigInt for exact integer arithmetic.
Please enter an integer between 0 and 1500.
F(n) — the n-th term
6,765
F(n−1) — the previous term
4,181
Consecutive ratio F(n) ÷ F(n−1)
1.61803399
Golden ratio φ ≈ 1.61803399
First 21 terms
Starting at F(0); up to the first 30 terms shown.
F(0)=0 and F(1)=1; every subsequent term is the sum of the previous two. The ratio of consecutive terms converges to the golden ratio φ ≈ 1.6180. Calculations use BigInt, so they remain exact past JavaScript's 53-bit floating-point limit.
Formula
F(0) = 0, F(1) = 1, F(n) = F(n−1) + F(n−2) Consecutive ratio: F(n) ÷ F(n−1) → φ = (1 + √5) ÷ 2 ≈ 1.6180339887…
- · Reference values: F(10) = 55, F(20) = 6,765, F(30) = 832,040, F(50) = 12,586,269,025, F(100) = 354,224,848,179,261,915,075. Source: OEIS sequence A000045.
- · The sequence was popularised in Europe by Leonardo Fibonacci's Liber Abaci (1202); modern references include OEIS A000045 and Vajda, "Fibonacci & Lucas Numbers" (Ellis Horwood, 1989).
- · The consecutive ratio F(n+1) ÷ F(n) oscillates above and below the golden ratio φ = (1 + √5) / 2 ≈ 1.6180339887 and converges to it; by n ≈ 20 the ratio matches φ to seven decimal places.
- · F(n) grows exponentially — each step multiplies by roughly φ — so the digit count grows by about one every five terms (log₁₀ φ ≈ 0.20898). F(1000) has 209 digits.
- · All computation uses JavaScript BigInt, so additions are exact past the 53-bit floating-point limit. The cap is set to n = 1500, which is more than enough for typical homework and number-theory practice.
- · Common variants include the Lucas numbers (L₀=2, L₁=1) and the Tribonacci sequence (sum of the previous three). This tool only outputs the standard Fibonacci sequence; you can derive Lucas numbers via L(n) = F(n−1) + F(n+1).
Frequently asked
What is F(100), and can JavaScript compute it accurately?
F(100) = 354,224,848,179,261,915,075 — 21 digits. Plain JavaScript Numbers carry only 53 bits of precision (Number.MAX_SAFE_INTEGER ≈ 9 × 10¹⁵, about 16 digits) and start losing precision from F(78) onward. This tool uses BigInt addition throughout, so every step is exact — even F(1500) returns instantly with no rounding.
Why does the ratio of consecutive Fibonacci numbers converge to the golden ratio?
Let r = lim F(n+1) ÷ F(n). Dividing both sides of the recurrence by F(n) gives F(n+1)/F(n) = 1 + F(n−1)/F(n), so r = 1 + 1/r, which rearranges to r² − r − 1 = 0. The positive root is r = (1 + √5) / 2 = φ ≈ 1.61803. The closed form (Binet's formula) F(n) = (φⁿ − ψⁿ) / √5 with ψ = 1 − φ ≈ −0.618 makes the same limit explicit.
Can I compute F(n) directly with Binet's closed-form formula instead of iterating?
In theory yes: F(n) = (φⁿ − ψⁿ) / √5. In practice √5 and φ are irrational, so floating-point evaluation introduces rounding error and loses accuracy from about n ≈ 70 onward. For exact integer results across all n you still need BigInt — either via the O(n) recurrence (what this tool uses) or matrix exponentiation. The plain recurrence handles n = 1500 in milliseconds.
Is 0 a Fibonacci number? Or does the sequence start at F(1) = 1?
It depends on the convention. Modern references — including OEIS A000045 — use F(0) = 0 and F(1) = 1, so 0 is a Fibonacci number. Some older textbooks index from F(1) = 1, F(2) = 1, skipping zero; aside from the starting index the two conventions agree. This tool follows the OEIS convention.
Related tools
Percentage Calculator
Percent of, percent change, and percent add/subtract in one.
GCD & LCM Calculator
Enter 2–6 positive integers to get the greatest common divisor (HCF / GCD) and least common multiple (LCM), with the Euclidean step chain shown.
Average Calculator (Mean / Median / Mode)
Enter a list of numbers to get the mean, median, mode, range plus standard deviation, variance and total.
Quadratic Equation Solver
Enter the coefficients of ax² + bx + c = 0 to find the real or complex roots, discriminant and vertex.
Password Strength (Entropy) Calculator
Estimate a password's bit entropy, brute-force time and strength tier. All computation happens in your browser.
Scientific Notation Converter
Convert between standard decimal numbers and scientific notation, with significant figures and order of magnitude.
Permutations & Combinations (nPr / nCr) Calculator
Compute permutations P(n,r), combinations C(n,r) and factorial n! — useful for probability problems, lottery odds and combinatorics homework.
Standard Deviation Calculator
Paste a list of numbers to compute mean, median, sample and population variance and standard deviation — with the working shown.
Triangle Calculator (SSS / SAS / ASA)
Solve a triangle from 3 sides, 2 sides + 1 angle, or 2 angles + 1 side — area, perimeter and remaining parts via the law of sines / cosines.
Pythagorean Theorem Calculator
Given any two sides of a right triangle (two legs, or one leg plus the hypotenuse), instantly find the third side, area, perimeter and the two non-right angles.
Circle Calculator (radius / diameter / circumference / area)
Enter any one of radius, diameter, circumference or area to get the other three — useful for design, engineering and DIY.
Roman Numeral Converter
Two-way conversion between Arabic numbers (1–3999) and Roman numerals (I, V, X, L, C, D, M) — handy for typesetting, chapter titles and homework.
Slope & Line Equation Calculator (y = mx + b from Two Points)
Enter two points (x₁, y₁) and (x₂, y₂) to instantly get the slope, y-intercept, line equation y = mx + b, distance, and midpoint — a classroom staple for algebra and coordinate geometry.
Birthday Paradox Calculator
Enter group size n to see the probability that at least two people share a birthday — the classic birthday problem.
Logarithm Calculator (log / ln / log₂ / any base)
Compute logₐ(x) for any base — natural log (ln), common log (log₁₀), binary log (log₂) and a custom base, with the change-of-base steps shown.
Z-Score (Standard Score) Calculator
Enter a value, the mean and the standard deviation to compute the z-score and the corresponding normal-distribution percentile and probabilities.
Screen Pixel Density (PPI) Calculator
Enter the screen resolution and diagonal size to get pixel density (PPI), real width/height, dot pitch and total pixel count.
Hong Kong Mark Six Odds Calculator
Enter the number of tickets / selections and see the actual probability of hitting first, second … prizes in a Mark Six (6-of-49) draw.
Decimal to Fraction Converter
Convert any decimal (including repeating decimals) to a simplified fraction and a mixed number.
Sphere Volume & Surface Area Calculator
Give a sphere any one of radius, diameter, surface area or volume and instantly get the other three — plus the great-circle circumference and area.
Cylinder Volume & Surface Area Calculator
Enter the radius and height of a cylinder to get volume (π r²h), lateral surface, base area and total surface area.
Prime Factorization Calculator
Factor any integer from 2 up to 10¹² into primes, see the canonical exponent form, count and list all divisors.
Geometric Mean Calculator
Compute the n-th root of the product of n positive numbers — the right average for growth rates, returns and ratios — alongside the arithmetic mean for comparison.