1. What is the Fibonacci Sequence?

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, starting from 0 and 1. The sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, etc.

2. How Does the Calculator Work?

The calculator uses the Fibonacci recurrence relation:

With base cases:

• \( F(0) = 0 \)
• \( F(1) = 1 \)

Explanation: The calculator computes Fibonacci numbers iteratively to efficiently handle larger values of n.

3. Importance of Fibonacci Numbers

Details: Fibonacci numbers appear in mathematics, computer science, biology, and art. They are used in algorithms, financial markets analysis, and appear in biological settings like branching in trees.

4. Using the Calculator

Tips: Enter a non-negative integer n (0-70) to calculate the nth Fibonacci number. The calculator uses arbitrary-precision arithmetic to handle large numbers accurately.

5. Frequently Asked Questions (FAQ)

Q1: Why is the limit set to n=70?
A: While the calculator can handle larger values, we limit to n=70 to prevent extremely large results that might not be practical for most uses.

Q2: How are negative numbers handled?
A: This calculator only computes Fibonacci numbers for non-negative integers (n ≥ 0).

Q3: What's the time complexity of this calculation?
A: The iterative method used has O(n) time complexity, making it efficient for the allowed range.

Q4: Can I calculate Fibonacci numbers recursively?
A: While possible, recursive calculation has O(2^n) time complexity and is impractical for n > 30.

Q5: What applications use Fibonacci numbers?
A: They're used in algorithms (Fibonacci heap, search techniques), financial analysis (Fibonacci retracements), and appear in nature (phyllotaxis).