1. What is the Binet Formula?

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

2. How Does the Calculator Work?

The calculator uses Binet's formula:

Where:

• \( F_n \) — The nth Fibonacci number
• \( \phi \) — Golden ratio (1.618...)
• \( n \) — The index of the Fibonacci number to calculate
• \( \cos(\pi n) \) — Alternates between -1 and 1 based on n

Explanation: The formula combines the golden ratio's powers with trigonometric functions to directly compute Fibonacci numbers.

3. Importance of Binet's Formula

Details: While not practical for exact integer calculations with large n (due to floating-point limitations), Binet's formula demonstrates the deep connection between Fibonacci numbers and the golden ratio, and is useful for theoretical analysis.

4. Using the Calculator

Tips: Enter n (a non-negative integer) and φ (golden ratio, default 1.618). The calculator will compute the nth Fibonacci number using Binet's formula.

5. Frequently Asked Questions (FAQ)

Q1: Why does Binet's formula work for Fibonacci numbers?
A: It's derived from the closed-form solution to the Fibonacci recurrence relation, using the golden ratio which appears naturally in the characteristic equation.

Q2: Is this exact for all integer n?
A: Mathematically yes, but in practice floating-point precision limits exactness for large n (n > 70 typically).

Q3: What's the role of the cosine term?
A: It handles the alternating sign in the formula's second term, equivalent to (-1)^n.

Q4: Can I use this for non-integer n?
A: The formula extends to real numbers, but Fibonacci numbers are only defined for integer indices.

Q5: How is this related to calculus?
A: The formula demonstrates how discrete sequences can be represented using continuous functions, bridging discrete and continuous mathematics.