Fibonacci Calculator
Calculate Fibonacci numbers and sequences
How to Use This Calculator
Enter Term Number
Input a non-negative integer (0, 1, 2, 3, ...) to find the nth Fibonacci number.
Click Calculate
Press the "Calculate" button to see the Fibonacci number and sequence.
Review Results
See the specific Fibonacci number and the complete sequence from F₀ to F_n.
Formula
Recursive: F₀ = 0, F₁ = 1, Fₙ = Fₙ₋₁ + Fₙ₋₂ for n ≥ 2
or
Binet's Formula: F_n = (φⁿ - ψⁿ) / √5
where φ = (1+√5)/2 ≈ 1.618 (golden ratio), ψ = (1-√5)/2
Example: Find F₁₀
F₀ = 0, F₁ = 1
F₂ = 0 + 1 = 1
F₃ = 1 + 1 = 2
F₄ = 1 + 2 = 3
...continuing...
F₁₀ = 55
About Fibonacci Calculator
The Fibonacci sequence is one of the most famous sequences in mathematics, named after Italian mathematician Leonardo Fibonacci. Each number in the sequence is the sum of the two preceding ones, starting from 0 and 1.
The Sequence
The Fibonacci sequence starts: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...
Key Properties
- Each number is the sum of the two preceding numbers
- As numbers increase, the ratio of consecutive Fibonacci numbers approaches the golden ratio (φ ≈ 1.618)
- Appears naturally in many phenomena: flower petals, spiral shells, pine cones
- Used in algorithms, computer science, and mathematical modeling
Applications
- Computer algorithms (Fibonacci heap, search techniques)
- Financial modeling (Fibonacci retracements in trading)
- Art and design (golden ratio applications)
- Nature study (phyllotaxis, spiral patterns)
Frequently Asked Questions
What are the first few Fibonacci numbers?
F₀ = 0, F₁ = 1, F₂ = 1, F₃ = 2, F₄ = 3, F₅ = 5, F₆ = 8, F₇ = 13, F₈ = 21, F₉ = 34, F₁₀ = 55
Why does F₀ = 0?
The Fibonacci sequence can start with F₀ = 0 or F₁ = 1, depending on convention. Modern definitions often use F₀ = 0 for mathematical consistency.
What is the golden ratio connection?
As n increases, F_1/F_n approaches the golden ratio φ ≈ 1.618. This ratio appears throughout nature and art.
Can I calculate large Fibonacci numbers?
Yes! The calculator can handle large values, though very large numbers may take longer to compute.
Are there negative Fibonacci numbers?
The Fibonacci sequence can be extended to negative indices using F_0 = (-1)^1 F_n, but this calculator focuses on non-negative indices.