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 appears in many natural phenomena and has important applications in mathematics and computer science.

2. How Does the Calculator Work?

The calculator uses the Fibonacci recurrence relation:

With initial conditions:

• \( F_0 = 0 \)
• \( F_1 = 1 \)

Explanation: The calculator computes Fibonacci numbers iteratively for efficiency, especially important for larger values of n.

3. Importance of Fibonacci Numbers

Details: Fibonacci numbers appear in biological settings, computer algorithms, financial markets, and have connections to the golden ratio. They are fundamental in algorithm analysis and recursive programming.

4. Using the Calculator

Tips: Enter a non-negative integer n to calculate the nth Fibonacci number. For large n (>100), computation may take longer.

5. Frequently Asked Questions (FAQ)

Q1: What's the largest Fibonacci number this can calculate?
A: It depends on your server's configuration, but typically up to n=1000 or more before hitting integer size limits.

Q2: Why is F₀ = 0?
A: This is the modern convention, though some older definitions start with F₁ = F₂ = 1.

Q3: How are Fibonacci numbers used in real life?
A: They appear in phyllotaxis (leaf arrangements), computer algorithms, financial analysis, and art/composition.

Q4: What's the time complexity of this calculation?
A: The iterative implementation runs in O(n) time, which is optimal for this calculation.

Q5: Can this calculate negative Fibonacci numbers?
A: No, this calculator only handles non-negative integers. The Fibonacci sequence can be extended to negative indices, but that's not implemented here.