Fibonacci Calculator
Calculate Fibonacci sequence
About This Tool
The Fibonacci Calculator generates Fibonacci numbers and sequences. The Fibonacci sequence is a series where each number is the sum of the two preceding ones, starting from 0 and 1.
Find any Fibonacci number, generate sequences of any length, and explore the fascinating connection between Fibonacci numbers and the golden ratio.
How It Works
How to Use This Calculator
Working with Fibonacci numbers is easy:
- Find nth Fibonacci - Enter n to find the nth Fibonacci number.
- Generate sequence - Specify how many terms to generate.
- Check if Fibonacci - Enter a number to see if it's in the sequence.
- Explore ratios - See how consecutive Fibonacci numbers approach the golden ratio.
Formula
Fibonacci Formula
Recursive definition:
- F(0) = 0
- F(1) = 1
- F(n) = F(n-1) + F(n-2) for n > 1
Binet's Formula (closed form): F(n) = (phi^n - psi^n) / sqrt(5) where phi = (1 + sqrt(5)) / 2 (golden ratio) and psi = (1 - sqrt(5)) / 2
First 15 Fibonacci numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377
Examples
Examples
Example 1: Find nth Fibonacci
- Input: n = 10
- Result: F(10) = 55
- Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
Example 2: Generate Sequence
- Input: First 8 terms
- Result: 0, 1, 1, 2, 3, 5, 8, 13
Example 3: Golden Ratio
- F(20)/F(19) = 6765/4181 = 1.61803
- Golden ratio phi = 1.61803
Frequently Asked Questions
What is the Fibonacci sequence?
The Fibonacci sequence starts with 0 and 1, and each subsequent number is the sum of the two preceding ones: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144... The sequence was introduced by Leonardo of Pisa in 1202.
How is the Fibonacci sequence related to the golden ratio?
The ratio of consecutive Fibonacci numbers converges to the golden ratio φ ≈ 1.6180339887. For example, 89/55 = 1.6181... and 144/89 = 1.6179... The larger the numbers, the closer the ratio gets to φ.
Where does the Fibonacci sequence appear in nature?
Fibonacci numbers appear in sunflower seed spirals (usually 34 and 55 spirals), pinecone scales, pineapple eyes, leaf arrangements (phyllotaxis), and shell spirals. These patterns optimize packing efficiency and light exposure.
How is Fibonacci used in trading and finance?
Fibonacci retracement levels (23.6%, 38.2%, 50%, 61.8%, 78.6%) are used in technical analysis to identify potential support and resistance levels. Traders use these ratios to predict price pullback targets during trends.
What is the formula for the nth Fibonacci number?
Binet's formula gives the nth Fibonacci number directly: F(n) = (φ^n - ψ^n) / √5, where φ = (1+√5)/2 and ψ = (1-√5)/2. For large n, F(n) ≈ φ^n / √5 since ψ^n approaches zero.
Disclaimer
Disclaimer: This calculator provides results for general reference and educational purposes only. While we strive for accuracy, results for very large numbers may be subject to computational limitations. All processing happens in your browser.
💡 Tips
Tips
- Fibonacci numbers grow exponentially
- The ratio of consecutive Fibonacci numbers approaches phi (golden ratio)
- Every 3rd Fibonacci number is even
- Fibonacci appears in nature: flower petals, pinecones, shells
- Used in algorithms, trading, and art