Fibonacci Sequence
IT 위키
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.
1 Definition[편집 | 원본 편집]
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
2 Example Sequence[편집 | 원본 편집]
The first few Fibonacci numbers are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...
3 Closed-Form Formula (Binet's Formula)[편집 | 원본 편집]
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
4 Properties[편집 | 원본 편집]
- 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.
5 Computational Methods[편집 | 원본 편집]
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.
6 Applications[편집 | 원본 편집]
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.
7 Comparison of Computational Methods[편집 | 원본 편집]
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) |