1. What is the Binet Formula?

The Binet formula is a closed-form expression for finding Fibonacci numbers without recursion. It uses the golden ratio (φ ≈ 1.618) to directly compute the nth Fibonacci number.

2. How Does the Calculator Work?

The calculator uses Binet's formula:

Where:

• \( F_n \) — The nth Fibonacci number
• \( \phi \) — Golden ratio (1.6180339887...)
• \( n \) — Term number (non-negative integer)

Explanation: The formula provides an exact integer result for Fibonacci numbers by leveraging the properties of the golden ratio and its conjugate.

3. Importance of Binet's Formula

Details: While recursive methods require O(n) time, Binet's formula allows for O(1) time calculation of Fibonacci numbers, making it valuable for large n values.

4. Using the Calculator

Tips: Enter any non-negative integer n to calculate the nth Fibonacci number. The calculator handles values up to n=70 accurately with standard floating-point precision.

5. Frequently Asked Questions (FAQ)

Q1: Why does Binet's formula work for integers?
A: The irrational parts cancel out exactly, leaving only the integer Fibonacci number.

Q2: What are the limitations of this formula?
A: Due to floating-point precision limits, direct computation becomes inaccurate for very large n (>70 with standard double precision).

Q3: How is this related to the golden ratio?
A: The ratio of consecutive Fibonacci numbers approaches φ as n increases, demonstrating the deep connection between Fibonacci numbers and the golden ratio.

Q4: Can this formula be used for negative n?
A: Yes, with F_-n = (-1)ⁿ⁺¹F_n, though this calculator focuses on non-negative n.

Q5: What's the largest n this calculator can handle?
A: Up to n=70 gives exact results; beyond that, floating-point rounding errors may occur.