Fibonacci Calculator
Generate Fibonacci sequence terms and explore the golden ratio relationship.
Single Term Result
F =
Fibonacci Sequence
Golden Ratio Approximation
Fn / Fn-1 =
Golden Ratio (φ) ≈ 1.6180339887...
Difference:
About the Fibonacci Sequence
The Fibonacci sequence is one of the most famous sequences in mathematics. It starts with 0 and 1, and each subsequent number is the sum of the two preceding ones. This sequence appears in many areas of mathematics and science, from number theory to biological patterns.
Definition of the Fibonacci Sequence
The Fibonacci sequence (Fn) is defined by the recurrence relation:
F0 = 0
F1 = 1
Fn = Fn-1 + Fn-2 for n > 1
This gives us the sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...
Binet's Formula
For larger Fibonacci numbers, we can use Binet's formula:
Fn = (φn - (1-φ)n)/√5
Where φ (phi) is the golden ratio, approximately equal to 1.6180339887...
The Golden Ratio Connection
As the Fibonacci sequence progresses, the ratio of consecutive terms (Fn / Fn-1) approaches the golden ratio (φ), which is approximately 1.6180339887... This ratio is found throughout nature, art, and architecture, and is considered aesthetically pleasing to the human eye.
Applications of the Fibonacci Sequence
- Nature: The pattern appears in the arrangement of leaves on stems, the spirals of shells, and the family trees of honeybees.
- Finance: Fibonacci retracements are used in technical analysis to predict potential reversal points in financial markets.
- Computer Science: Fibonacci numbers are used in the analysis of algorithms and data structures.
- Art and Architecture: The golden ratio derived from Fibonacci numbers is used in design for balanced and aesthetically pleasing compositions.
- Music: Fibonacci sequences can be found in the structures of musical compositions.
Interesting Fibonacci Properties
- The sum of the first n Fibonacci numbers equals Fn+2 - 1
- The sum of the squares of the first n Fibonacci numbers equals Fn × Fn+1
- Every third Fibonacci number is divisible by 2
- Every fourth Fibonacci number is divisible by 3
- Every fifth Fibonacci number is divisible by 5
- The greatest common divisor of two Fibonacci numbers Fm and Fn is Fgcd(m,n)
How to Use This Calculator
- To calculate a specific Fibonacci term, enter the term number (n) and click "Calculate Term"
- To generate a sequence of Fibonacci numbers, enter the desired length and click "Generate Sequence"
- The calculator will also show the golden ratio approximation for higher terms
Whether you're a student learning about this fascinating sequence, a mathematician exploring its properties, or someone applying Fibonacci numbers in practical applications, our calculator provides a simple way to work with Fibonacci numbers.
Frequently Asked Questions
How large can the Fibonacci numbers get?
Fibonacci numbers grow exponentially at a rate close to the golden ratio (approximately 1.618). For example, the 50th Fibonacci number is 12,586,269,025, and the 100th is 354,224,848,179,261,915,075. Our calculator can compute these large values efficiently, but for extremely large terms (beyond 1,476), JavaScript's number precision becomes a limitation.
Is 0 considered the first Fibonacci number?
There are two common conventions. The modern definition typically starts with F0 = 0 and F1 = 1, making 0 the first term (zeroth index). Some older sources start with F1 = 1 and F2 = 1. Our calculator uses the modern convention where F0 = 0.
Why is the Fibonacci sequence important in nature?
The Fibonacci sequence and golden ratio appear in numerous natural phenomena. They can be observed in the spiral arrangements of leaves, pine cones, and sunflower seeds; the branching patterns of trees; the shell spirals of mollusks; and even the proportions of the human body. These patterns often represent efficient solutions for growth and resource allocation in biological systems.
How is the Fibonacci sequence related to the golden spiral?
The golden spiral is created by drawing quarter-circle arcs connecting opposite corners of squares with sides equal to successive Fibonacci numbers. As the Fibonacci numbers grow, the ratio between consecutive numbers approaches the golden ratio, and the resulting spiral approaches the golden spiral, which is based directly on the golden ratio. This spiral appears in nature in phenomena like nautilus shells and hurricane formations.
Can Fibonacci numbers be negative?
Yes, the Fibonacci sequence can be extended to negative indices. The formula remains Fn = Fn-1 + Fn-2, but now we compute backwards. For negative indices, the pattern follows: F-1 = 1, F-2 = -1, F-3 = 2, F-4 = -3, and so on. This extension has the property that F-n = (-1)n+1 × Fn.
How are Fibonacci numbers used in computer algorithms?
Fibonacci numbers appear in many algorithms and data structures. The Fibonacci search technique is used to find elements in sorted arrays. Fibonacci heaps are used in efficient graph algorithms like Dijkstra's shortest path. The sequence is also used in pseudorandom number generators, and the Fibonacci tree structure illustrates recursive algorithms' behavior and provides worst-case scenarios for certain balanced tree operations.