Fibonacci Sequence Calculator
Calculate n-th Fibonacci number, sum of terms, and generate complete sequences
Which Fibonacci number to find (0-indexed: F₀=0, F₁=1, F₂=1...)
How many terms to include (starting from F₀)
Classic Fibonacci uses F₀=0, F₁=1. Try F₀=2, F₁=1 for Lucas numbers.
Quick Examples
Fibonacci Number F() Sum of F(0) to F() Fibonacci Sequence
Calculation Details
n-th Term
F()
Sum of Terms
Σ F(0)...F()
Terms Generated
F(0) to F()
Ratio (≈ φ)
Famous Fibonacci Numbers
| Position (n) | F(n) | Notes |
|---|---|---|
| 0 | 0 | First term |
| 1 | 1 | Second term |
| 10 | 55 | Common example |
| 12 | 144 | Only Fibonacci number that is a perfect square (besides 0 and 1) |
| 20 | 6,765 | Ratio to F(19) ≈ 1.6180339... |
| 30 | 832,040 | First Fibonacci > 800,000 |
| 50 | 12,586,269,025 | Over 12 billion |
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About Fibonacci Sequence Calculator
What is the Fibonacci Sequence?
The Fibonacci sequence is one of the most famous sequences in mathematics. Each number in the sequence is the sum of the two preceding numbers, typically starting with 0 and 1.
Classic Fibonacci Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144...
The sequence was introduced to Western mathematics by Leonardo of Pisa (known as Fibonacci) in his 1202 book Liber Abaci.
How to Use This Calculator
- Select a calculation mode - Choose whether to find the n-th term, calculate the sum, or generate a sequence
- Enter n - The position or number of terms you want
- Optional: Customize starting values - Use the advanced options to start with different values (F₀ and F₁)
- View results - See the calculated value with step-by-step explanation
Key Formulas
Recursive Definition
F(n) = F(n-1) + F(n-2)
Where:
- F(0) = 0 (first term)
- F(1) = 1 (second term)
- F(n) = the n-th Fibonacci number
Binet's Formula (Closed Form)
F(n) = (φⁿ - ψⁿ) / √5
Where:
- φ (phi) = (1 + √5) / 2 ≈ 1.618034 (Golden Ratio)
- ψ (psi) = (1 - √5) / 2 ≈ -0.618034
Sum Formula
∑F(i) from i=0 to n = F(n+2) - 1
The sum of the first n+1 Fibonacci numbers equals F(n+2) - 1.
The Golden Ratio Connection
The ratio of consecutive Fibonacci numbers approaches the Golden Ratio (φ ≈ 1.618034) as n increases:
lim(n→∞) F(n+1)/F(n) = φ
This connection makes Fibonacci numbers appear throughout nature, art, and architecture.
Applications
| Field | Application |
|---|---|
| Nature | Flower petals, seed arrangements, spiral shells |
| Computer Science | Algorithm analysis, data structures |
| Finance | Technical analysis, Fibonacci retracement |
| Art & Design | Golden ratio compositions |
| Biology | Population growth models |
Examples
Finding the 10th Fibonacci Number
F(10) = 55
Sequence: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55
Sum of First 7 Fibonacci Numbers
0 + 1 + 1 + 2 + 3 + 5 + 8 = 20
Using the formula: F(9) - 1 = 34 - 1 = 33 ✗ Note: Sum of F(0) to F(6) = F(8) - 1 = 21 - 1 = 20 ✓
Frequently Asked Questions
Why is Fibonacci sequence important?
The Fibonacci sequence appears in numerous natural phenomena, from the spiral arrangement of leaves to the breeding patterns of rabbits. It's also fundamental in computer science algorithms and financial market analysis.
What is the relationship with the Golden Ratio?
As Fibonacci numbers get larger, the ratio between consecutive numbers approaches the Golden Ratio (1.618034...). This ratio appears throughout nature and is considered aesthetically pleasing in art and architecture.
Can Fibonacci sequence start with different numbers?
Yes! While the classic sequence starts with 0 and 1, you can start with any two numbers. The Lucas sequence, for example, starts with 2 and 1: 2, 1, 3, 4, 7, 11, 18...
How fast do Fibonacci numbers grow?
Fibonacci numbers grow exponentially, approximately by a factor of φ (1.618) for each step. By the 50th term, the number exceeds 12 billion.
Fun Fact: The Fibonacci sequence was originally described to model the idealized reproduction of rabbits!