Generate the Fibonacci sequence, the nth term and the running sum.
Calculated locally in your browser.
What is the Fibonacci sequence and how is it generated?
It starts F₀ = 0, F₁ = 1, and each term after is the sum of the two before: Fₙ = Fₙ₋₁ + Fₙ₋₂. The ratio of consecutive terms approaches the golden ratio φ ≈ 1.618. The first 10 terms are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, which sum to 88, and 34 ÷ 21 ≈ 1.619.
Understanding your result
Each Fibonacci number is the sum of the previous two, so the sequence grows quickly. The ratio of one term to the previous gets closer and closer to the golden ratio φ ≈ 1.618. Terms are exact up to F(78); beyond that, standard numbers lose integer precision.
Formula and method
The sequence starts F₀ = 0, F₁ = 1, and each term after is the sum of the two before it: Fₙ = Fₙ₋₁ + Fₙ₋₂.
Worked example
The first 10 terms are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34 — they sum to 88, and 34 ÷ 21 ≈ 1.619, close to the golden ratio.
How to use this tool
- Enter how many terms you want.
- Press Generate.
- Read the sequence, nth term and sum.
Common mistakes to avoid
- Disagreeing on the start — some lists begin 1, 1 instead of 0, 1.
- Expecting exact values for very large terms.
- Confusing the term index with the term value.
About the Fibonacci Calculator
The Fibonacci Calculator generates the Fibonacci sequence to any number of terms and shows the nth term, the sum of the terms and how the ratio of consecutive terms approaches the golden ratio.
Who should use this tool
Students, teachers, programmers and anyone curious about the famous sequence.
Benefits
- The full sequence to your chosen length.
- The nth term and the running sum.
- Shows the ratio converging to the golden ratio φ.
- Private — calculated entirely in your browser.
Practical use cases
- Listing Fibonacci numbers for a maths exercise.
- Finding a specific term in the sequence.
- Demonstrating the golden ratio in class.
Frequently asked questions
Where does the Fibonacci sequence start?
This calculator starts at 0, 1 (F₀ = 0, F₁ = 1). Some sources start at 1, 1; the numbers are otherwise the same.
How is it related to the golden ratio?
As the sequence grows, the ratio of consecutive terms approaches the golden ratio φ ≈ 1.618.
Why are very large terms approximate?
Beyond F(78) the values exceed the largest exact integer a standard number can hold, so they become approximate.