What is the 100th Fibonacci Number?

What is the Fibonacci sequence?

The Fibonacci sequence starts with 0 and 1. Every number after that is the sum of the two before it: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, and so on. This pattern, first described in Western mathematics by Leonardo of Pisa in 1202, appears in sunflower seed spirals, pine cone scales, shell growth, branching patterns in trees, and even in financial market analysis. This calculator lets you find any term F(n) up to n = 1000, generate a table of the first N terms with golden ratio convergence, or test whether a given number belongs to the sequence.

Quick start example

Want to know the 10th Fibonacci number? Select Nth term of the sequence, enter 10, click Calculate.
F(10) = 55
The calculator also shows that F(10)/F(9) = 1.6176..., already close to the golden ratio 1.6180339887.

Key formulas and properties

Property	Formula / Value	Notes
Recursive definition	F(n) = F(n-1) + F(n-2)	F(0) = 0, F(1) = 1
Golden ratio (phi)	(1 + sqrt(5)) / 2 = 1.6180339887...	F(n)/F(n-1) converges to phi
Binet's formula	F(n) = (phi^n - psi^n) / sqrt(5)	psi = (1 - sqrt(5))/2
Digit count	F(n) has roughly n * log10(phi) digits	F(1000) has 209 digits
Fibonacci check	n is Fibonacci iff 5n^2+4 or 5n^2-4 is a perfect square	Used by this calculator
Sum of first n terms	F(0)+F(1)+...+F(n) = F(n+2) - 1	Useful identity

Practical examples

Example 1 - small term
Find F(7). The sequence goes: 0, 1, 1, 2, 3, 5, 8, 13. So F(7) = 13. The calculator confirms this and shows the formula: F(7) = F(6) + F(5) = 8 + 5 = 13.

Example 2 - large term with golden ratio
Find F(50). The result is 12,586,269,025. The ratio F(50)/F(49) = 1.6180339887, matching the golden ratio to 10 decimal places. This convergence is remarkably fast: by F(12)/F(11) the ratio is already 1.6180555..., accurate to 4 decimal places.

Example 3 - very large term
F(500) is a number with 105 digits. The calculator uses arbitrary-precision arithmetic (BigInt) to compute it exactly. Even F(1000) with its 209 digits is computed in milliseconds.

Example 4 - list of first 20 terms
Select "List of first N terms" and enter 20. You get a table from F(0) = 0 to F(19) = 4,181 with the ratio column showing how quickly F(n)/F(n-1) converges to 1.618034. The sum of all 20 terms is also displayed.

Example 5 - checking a number
Is 144 a Fibonacci number? Select "Check if a number is Fibonacci", enter 144. The answer: YES, it is F(12). Try 150: NO, it is not a Fibonacci number. The nearest terms are F(12) = 144 and F(13) = 233.

Example 6 - Fibonacci in nature
Sunflowers typically have 34 or 55 spirals going in opposite directions. Both 34 and 55 are consecutive Fibonacci numbers (F(9) and F(10)). Pine cones show 8 and 13 spirals. This pattern arises from the growth angle of approximately 137.5 degrees, which is related to the golden ratio.

Example 7 - computer science application
Fibonacci numbers appear in algorithm analysis. A Fibonacci heap data structure has amortized O(1) insert time. The worst-case input for the Euclidean algorithm (GCD) uses consecutive Fibonacci numbers. Fibonacci coding is used in data compression and search algorithms.

How the calculator works

For computing F(n), the calculator uses an iterative approach starting from F(0) = 0 and F(1) = 1, building up to the requested term. JavaScript's BigInt type handles arbitrarily large integers, so even F(1000) is computed exactly. For the Fibonacci membership test, it uses the mathematical property that a non-negative integer m is a Fibonacci number if and only if 5m^2 + 4 or 5m^2 - 4 is a perfect square. This avoids generating the entire sequence just to check a single number.

FAQ

What is the golden ratio and how is it related to Fibonacci?

The golden ratio (phi) equals (1 + sqrt(5)) / 2, approximately 1.6180339887. As you go further in the Fibonacci sequence, the ratio F(n)/F(n-1) gets closer and closer to phi. By F(40)/F(39), the ratio matches phi to more than 15 decimal places. This relationship is not a coincidence: Binet's formula shows that F(n) is essentially phi^n divided by sqrt(5), rounded to the nearest integer.

Why does the sequence start with 0 and 1?

The modern convention sets F(0) = 0 and F(1) = 1. Some older textbooks start with F(1) = 1, F(2) = 1, omitting the initial zero. Both conventions produce the same sequence of numbers (0, 1, 1, 2, 3, 5, 8, ...) but differ in indexing. This calculator uses the modern convention because it leads to cleaner mathematical identities, such as gcd(F(m), F(n)) = F(gcd(m, n)).

How large a Fibonacci number can this calculator compute?

Up to F(1000), which has 209 digits. The calculator uses JavaScript BigInt for exact integer arithmetic with no floating-point rounding. F(1000) is computed in under a second. Higher values are capped to keep the page responsive, but the mathematical approach would work for even larger terms.

How does the Fibonacci membership test work?

A positive integer m is a Fibonacci number if and only if at least one of 5m^2 + 4 or 5m^2 - 4 is a perfect square. This is a proven mathematical theorem. The calculator computes both expressions using BigInt arithmetic and checks for perfect squares using Newton's method for integer square roots. This is far faster than generating the entire sequence.

Where do Fibonacci numbers appear in real life?

Fibonacci numbers appear in the spiral arrangement of seeds in sunflowers and pine cones, the branching of trees and ferns, the arrangement of leaves on a stem (phyllotaxis), the family tree of honeybees, and the proportions of the nautilus shell. In technology, they appear in Fibonacci heaps, Fibonacci search, Fibonacci coding, and as worst-case inputs for the Euclidean algorithm. In finance, Fibonacci retracement levels (23.6%, 38.2%, 61.8%) are derived from ratios of Fibonacci numbers.

What is the sum of the first N Fibonacci numbers?

There is an elegant identity: F(0) + F(1) + F(2) + ... + F(n) = F(n+2) - 1. For example, the sum of the first 10 terms (F(0) through F(9)) equals F(11) - 1 = 89 - 1 = 88. The calculator displays this sum when you generate a list of terms.

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100th Fibonacci number - calculate

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