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Fibonacci Sequence Formula:
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The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, starting from 0 and 1. It appears in many areas of mathematics and nature.
The calculator uses the Fibonacci recurrence relation:
With initial conditions:
Explanation: For any term n ≥ 2, the value is calculated by adding the two previous terms in the sequence.
Details: Fibonacci numbers appear in biological settings, computer algorithms, financial markets, and have connections to the golden ratio.
Tips: Enter a non-negative integer n to calculate the nth Fibonacci number. The sequence starts with F₀ = 0.
Q1: What are the first few Fibonacci numbers?
A: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...
Q2: Is there a closed-form formula for Fibonacci numbers?
A: Yes, Binet's formula: \( F_n = \frac{\phi^n - \psi^n}{\sqrt{5}} \) where ϕ is the golden ratio and ψ = 1 - ϕ.
Q3: What's the largest Fibonacci number this calculator can handle?
A: It depends on your system's integer size, but typically up to F₉₂ (7540113804746346429) for 64-bit systems.
Q4: Are there negative Fibonacci numbers?
A: Yes, the sequence can be extended to negative indices using \( F_{-n} = (-1)^{n+1}F_n \).
Q5: Where do Fibonacci numbers appear in nature?
A: In flower petal counts, pinecone spirals, sunflower seed arrangements, and many other biological patterns.