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 1 and 1. It appears in many areas of mathematics and nature.

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 (positive integer)

Explanation: The sequence begins with 1, 1, 2, 3, 5, 8, 13, 21, etc., where each subsequent number is the sum of the previous two.

3. Importance of Fibonacci Numbers

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

4. Using the Calculator

Tips: Enter a positive integer n to calculate the nth Fibonacci number. The calculator efficiently computes the result using an iterative approach.

5. Frequently Asked Questions (FAQ)

Q1: What is the largest Fibonacci number this calculator can compute?
A: It depends on your system's integer size, but typically up to F₉₂ (7540113804746346429) before overflow occurs with 64-bit integers.

Q2: Are there closed-form formulas for Fibonacci numbers?
A: Yes, Binet's formula provides an exact expression using the golden ratio, but it's less practical for exact integer computation.

Q3: How is the Fibonacci sequence related to the golden ratio?
A: The ratio of consecutive Fibonacci numbers approaches the golden ratio (1.618...) as n increases.

Q4: What are some real-world applications of Fibonacci numbers?
A: They appear in phyllotaxis (leaf arrangement), computer algorithms (Fibonacci heap), and technical analysis in trading.

Q5: Is there a Fibonacci sequence calculator for negative indices?
A: The Fibonacci sequence can be extended to negative integers using F_-n = (-1)ⁿ⁺¹F_n.