Fibonacci Sequence Generator
Generate Fibonacci numbers online. Calculate the nth Fibonacci number and view sequences with explanations.
About this calculator
Generate the first N Fibonacci numbers, or find a specific term F(n) directly. The page also shows the ratio of the last two terms and compares it with the golden ratio.
At 10 terms the ratio F(9)/F(8) is already 1.619, within 0.001 of φ.
How to use it
- Enter N and press Generate for the first N terms.
- Enter an index and press Find for a single term.
- Read the golden ratio estimate.
Rule
The Fibonacci sequence starts 0, 1 and each next term is the sum of the previous two: F(n) = F(n−1) + F(n−2). The ratio of consecutive terms approaches the golden ratio φ = (1 + √5) ÷ 2 ≈ 1.6180339887. Binet's formula gives a closed form: F(n) = (φⁿ − ψⁿ) ÷ √5 with ψ = (1 − √5) ÷ 2.
Worked example
First 10 terms
| Sequence | 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 |
| F(10) | 55 |
| F(9) / F(8) | 1.6190476190 |
| Golden ratio | 1.6180339887 |
Tips and common mistakes
- Use the nth-term finder for large indexes.
- The golden ratio is the limit of consecutive ratios.
- Fibonacci numbers appear in algorithms and finance (retracement levels).
- Check small values by hand.
- Indexing starts at F(0) = 0.
Frequently asked questions
What is the Fibonacci sequence?
A sequence where each number is the sum of the two before it: 0, 1, 1, 2, 3, 5, 8…
What is the 10th Fibonacci number?
F(10) is 55.
What is the golden ratio?
About 1.618, the limit of the ratio of consecutive Fibonacci numbers.
Where does it appear in nature?
In spirals of sunflowers, pine cones and shells, though sometimes loosely claimed.
Does it start from 0 or 1?
Conventions differ; this tool starts at F(0) = 0.
How fast does it grow?
Exponentially, by a factor of about 1.618 each step.