Fibonacci Sequence is a mathematical sequence where each number is the sum of the two preceding numbers. It is widely used in mathematics, computer science, and nature to model growth patterns and recursive structures.
Definition
The Fibonacci sequence is defined recursively as:
- F(0) = 0, F(1) = 1 (Base cases)
- F(n) = F(n-1) + F(n-2) for n ≥ 2
Example Sequence
The first few Fibonacci numbers are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
Closed-Form Formula (Binet's Formula)
The nth Fibonacci number can be computed directly using Binet’s formula:
- F(n) = (φⁿ - ψⁿ) / √5
where:
- φ = (1 + √5) / 2 (Golden Ratio)
- ψ = (1 - √5) / 2
Properties
- Recurrence Relation: Each term is the sum of the previous two.
- Golden Ratio Connection: The ratio of consecutive Fibonacci numbers converges to φ.
- Parity Pattern: Even numbers appear at every third position.
- Sum of Fibonacci Numbers: The sum of the first n Fibonacci numbers is F(n+2) - 1.
Computational Methods
There are several ways to compute Fibonacci numbers:
- Recursive Approach:
- Uses the recurrence relation directly.
- Complexity: O(2ⁿ) (Exponential, inefficient for large n).
- Iterative Approach:
- Uses a loop to compute Fibonacci numbers.
- Complexity: O(n) (Linear, more efficient than recursion).
- Matrix Exponentiation:
- Uses matrix multiplication to compute Fibonacci numbers efficiently.
- Complexity: O(log n).
- Binet’s Formula:
- Uses the closed-form expression.
- Complexity: O(1), but limited by floating-point precision.
Applications
Fibonacci numbers appear in various fields:
- Computer Science: Recursive algorithm analysis, dynamic programming.
- Mathematics: Number theory, combinatorial counting.
- Nature: Leaf arrangements, branching patterns in plants.
- Financial Markets: Fibonacci retracement in technical analysis.
- Art and Architecture: Proportions related to the golden ratio.
Comparison of Computational Methods
| Method | Approach | Complexity |
|---|---|---|
| Recursion | Direct recursive relation | O(2ⁿ) |
| Iteration | Loop-based calculation | O(n) |
| Matrix Exponentiation | Fast exponentiation of Fibonacci matrix | O(log n) |
| Binet’s Formula | Direct computation using golden ratio | O(1) |