Definition
A sequence where each number is the sum of the two preceding ones, often starting with 0 and 1.
Origin
First described in Indian mathematics around 200 BC by Pingala.
Popularization
Introduced to Western European mathematics by Leonardo of Pisa (Fibonacci) in 1202.
Applications
Used in computer algorithms, data structures, graphs, and observed in various biological settings.
Relation to Golden Ratio
Closely related to the golden ratio, with the ratio of two consecutive numbers tending to it as the sequence progresses.
Related to Lucas Numbers
Associated with Lucas numbers, sharing a similar recurrence relation and forming complementary Lucas sequences.
The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones. It typically starts with 0 and 1, leading to the sequence:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, and so forth.
The mathematical rule governing this sequence can be expressed as:
F(n) = F(n−1) + F(n−2) where F(n) is the nth Fibonacci number. To find out nth Fibonacci number in Mathematica, just type Fibonacci[n].
Historical Context
The sequence is named after the Italian mathematician Leonardo of Pisa, known as Fibonacci, who introduced it to the Western world in his 1202 work, Liber Abaci. However, the sequence itself was known in Indian mathematics long before Fibonacci’s time, appearing in ancient Sanskrit texts.
Mathematical Properties
The Fibonacci sequence exhibits several interesting properties:
Golden Ratio: The ratio of successive Fibonacci numbers approximates the golden ratio (approximately 1.618033988749895). As the sequence progresses, the ratio F(n)/F(n-1) approaches this value.
Occurrences in Nature: Fibonacci numbers frequently appear in biological settings, such as in the arrangement of leaves, the branching of trees, and the patterns of various fruits and flowers. For example, many flowers have a number of petals that is a Fibonacci number.
Applications in Finance: The Fibonacci sequence is also utilized in technical analysis in finance, where traders use Fibonacci retracement levels to predict potential reversal points in asset prices.
