Fibonacci Sequence Terms Calculator
List the first n Fibonacci numbers, with the nth term, the running sum, and the ratio that approaches the golden ratio.
How it works
F₁ = 0, F₂ = 1, Fₙ = Fₙ₋₁ + Fₙ₋₂ · Sum of first n terms = Fₙ₊₂ − 1 · Fₙ / Fₙ₋₁ → φ ≈ 1.618
The Fibonacci sequence starts 0, 1 and builds each term by adding the two before it, giving 0, 1, 1, 2, 3, 5, 8, 13 and onward. The ratio of consecutive terms converges to the golden ratio φ ≈ 1.618, and the running sum of the first n terms equals the (n+2)th term minus one. This calculator lists the terms up to a capped 60 so the numbers stay exact.
Worked example
The first 10 terms are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34. The 10th term is 34, their sum is 88 (which is F₁₂ − 1 = 89 − 1), and 34 ÷ 21 ≈ 1.619, already close to φ.
Sources: Fibonacci sequence — Wikipedia
Frequently asked questions
Where does the sequence start?
This calculator uses the common convention 0, 1, 1, 2, 3, 5, … so the first term is 0 and the second is 1. Some texts start at 1, 1; if you need that, read the term after the leading 0.
Why is the number of terms capped?
Fibonacci numbers grow exponentially, and beyond roughly the 60th term they exceed the range where ordinary floating-point numbers stay exact. Capping at 60 keeps every listed value precise.
How does Fibonacci relate to the golden ratio?
Dividing each term by the one before it produces a sequence that homes in on φ = (1+√5)/2 ≈ 1.618034 — the further along you go, the closer the ratio gets.
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This calculator is for educational purposes. Double-check important results before acting on them.