How this calculation works
Start at the explicitly numbered seed terms zero and one. Each next term adds the previous two. An iterative whole-number calculation keeps every digit exact and provides the next term and a checkable sum identity.
F₀ = 0, F₁ = 1, Fₙ = Fₙ₋₁ + Fₙ₋₂; Σₖ₌₀ⁿFₖ = Fₙ₊₂−1A worked example
With F₀ = 0 and F₁ = 1, F₁₀ = 55. The sum F₀ through F₁₀ is F₁₂ − 1 = 143.
Good to know
Indexes 0 through 500 are supported. The early-term table stops at index 20; the selected term and sum remain exact. This is a sequence calculation, not a claim that a real-world pattern follows Fibonacci growth.
Reference: OpenStax: Fibonacci numbers as a recursive sequence (opens in a new tab) ↗
Indexing begins at zero here
This page uses F₀=0 and F₁=1. Then F₂=1, F₃=2 and F₁₀=55. Some presentations start their displayed list at the first 1; checking the explicit index prevents a one-place disagreement.
The table shows early terms so the recurrence can be followed by eye. Even if you ask for index 500, the exact requested value is calculated; the table stops after index 20 to avoid hundreds of rows.
The arithmetic sequence calculator uses a constant difference; Fibonacci instead adds the previous two indexed terms.
A sum identity gives a second check
Adding F₀ through F₁₀ gives 143. The identity ΣFₖ from k=0 to n equals Fₙ₊₂−1, so here F₁₂−1=144−1=143. Showing the selected term, next term and sum lets you inspect more than one property of the recurrence.
The arithmetic sequence calculator uses a constant difference, and the geometric sequence calculator uses a constant ratio. Fibonacci uses the sum of the two preceding terms, so it is neither of those simple sequence models.
Exact integers matter at larger indexes
A floating-point closed formula can round large Fibonacci values. This tool adds whole numbers exactly and displays them as text when needed, preserving every digit through its supported index 500.
Fibonacci patterns occur in mathematical models, but a real population, price or natural shape does not automatically obey this recurrence. This page supplies the sequence, not an empirical growth forecast.
A few questions, answered
Is F₁₀ the tenth displayed row?
With F₀ shown as the first row, F₁₀ is the row at index ten.
Why does the table stop at twenty?
It keeps the explanation readable; the selected exact term and sum are still calculated up to index 500.
Does the ratio of two neighboring terms always equal the golden ratio?
No. The ratios vary; they approach that value as the index grows.