1. What is the Fibonacci Series?

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 closed-form formula (Binet's formula) allows direct calculation of any term without computing previous terms.

2. How Does the Formula Work?

The calculator uses Binet's formula:

Where:

• \( F_n \) — The nth Fibonacci number (integer)
• \( n \) — Term index (non-negative integer)
• \( \phi \) — Golden ratio (≈1.61803, unitless)

Explanation: This closed-form solution calculates Fibonacci numbers directly using the golden ratio, avoiding recursive computation.

3. Importance of Fibonacci Numbers

Details: Fibonacci numbers appear in mathematics, computer science, biology, and art. They model growth patterns in nature and have applications in algorithms and financial markets.

4. Using the Calculator

Tips: Enter a non-negative integer n to calculate the nth Fibonacci number. The golden ratio φ is also displayed for reference.

5. Frequently Asked Questions (FAQ)

Q1: Why does the formula give exact integers despite using irrational numbers?
A: The irrational parts cancel out, leaving exact integers due to the mathematical properties of the golden ratio.

Q2: What's the largest Fibonacci number this calculator can handle?
A: It depends on PHP's number precision, but typically accurate up to F₇₀ (≈1.2e14) with standard floating-point arithmetic.

Q3: Is this formula practical for computing large Fibonacci numbers?
A: For very large n, exact integer methods may be better due to floating-point precision limits, but this formula is mathematically elegant.

Q4: How is the golden ratio related to Fibonacci numbers?
A: The ratio of consecutive Fibonacci numbers approaches φ as n increases, demonstrating the deep connection between them.

Q5: Can this formula compute Fibonacci numbers for negative indices?
A: Yes, with F_-n = (-1)ⁿ⁺¹F_n, though this calculator focuses on non-negative indices.