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 begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on.

2. How Does the Calculator Work?

The calculator uses the Fibonacci recurrence relation:

Where:

• \( F(n) \) — The nth Fibonacci number
• \( n \) — Term position in the sequence (non-negative integer)

Explanation: Each number in the sequence is calculated by adding the two numbers before it.

3. Mathematical Properties

Details: The Fibonacci sequence appears in biological settings, computer algorithms, and has connections to the golden ratio. It grows exponentially, with the ratio of consecutive terms approaching the golden ratio (≈1.618).

4. Using the Calculator

Tips: Enter a non-negative integer (0-100) to calculate the nth Fibonacci number and see the sequence up to that term.

5. Frequently Asked Questions (FAQ)

Q1: Why does the sequence start with 0 and 1?
A: This is the modern convention, though some older definitions start with 1 and 1. Our calculator follows the 0-based indexing standard.

Q2: What's the largest Fibonacci number this calculator can compute?
A: The calculator is limited to n=100 (F(100) = 354224848179261915075) to prevent excessive computation.

Q3: Are there closed-form formulas for Fibonacci numbers?
A: Yes, Binet's formula provides an exact expression using the golden ratio, though it's less practical for computation than the iterative method.

Q4: What are some applications of Fibonacci numbers?
A: They appear in algorithm analysis, financial markets, biological growth patterns, and art/architecture.

Q5: How is this related to the golden ratio?
A: The ratio of consecutive Fibonacci numbers approaches the golden ratio (φ ≈ 1.618) as n increases.