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. Binet's formula provides an exact closed-form expression for the nth Fibonacci number using the golden ratio.

2. How Does the Calculator Work?

The calculator uses Binet's formula:

Where:

• \( F_n \) — The nth Fibonacci number
• \( \phi \) — Golden ratio (≈1.61803)
• \( n \) — Term number (0-based index)

Explanation: This formula allows direct computation of any Fibonacci number without needing to compute all previous terms.

3. Importance of Fibonacci Numbers

Details: Fibonacci numbers appear in biological settings, computer algorithms, financial markets, and have connections to the golden ratio in art and architecture.

4. Using the Calculator

Tips: Enter a non-negative integer n to calculate the nth Fibonacci number (F₀ = 0, F₁ = 1, F₂ = 1, etc.).

5. Frequently Asked Questions (FAQ)

Q1: Why does Binet's formula work for integers when it has √5?
A: The irrational parts cancel out, leaving an exact integer result for all non-negative integers n.

Q2: What's the largest Fibonacci number this can calculate?
A: Depends on PHP's number handling, but typically accurate up to F₇₈ before precision limits.

Q3: Is this more efficient than iterative calculation?
A: For single terms, yes. For multiple terms, iterative methods may be better.

Q4: How is this related to the golden ratio?
A: The ratio of consecutive Fibonacci numbers approaches φ as n increases.

Q5: Can this calculate negative-index Fibonacci numbers?
A: The formula works for negative n, producing the "negafibonacci" sequence.