First ten terms
\(0, 1, 1, 2, 3, 5, 8, 13, 21, 34\)
Fibonacci Sequence Calculator
Calculate Fibonacci numbers with golden ratio connection and sequence visualization
Explore the Fibonacci sequence
Drag n to grow the sequence. The bar chart shows F₀ through F_n; the ratio F_n / F_{n-1} converges on the golden ratio φ.
Quick presets
10
050
Predict what will happen
What does F_n / F_{n-1} approach as n grows?
Watch the callout as you slide n from 5 to 30.
Golden ratio
The ratio of consecutive Fibonacci numbers approaches φ ≈ 1.618034. φ is the positive root of x² = x + 1 — the same recurrence the sequence follows.
Common mistake
F_0 = 0 and F_1 = 1. Some sources start at F_1 = 1, F_2 = 1; double-check the indexing convention before comparing answers.
Why it works
Each term is the sum of the two before it: a doubly-recursive definition. Iteratively summing forward avoids the exponential blow-up of the naive recursion.
Results
Final Answer
\(F_{10} = 55\)
Step-by-step Solution
- Find Fibonacci term \(F_{10}\)
- Recursive formula \(F_n = F_{n-1} + F_{n-2},\quad F_0 = 0,\, F_1 = 1\)
- Result \(F_{10} = 55\)
calculators.combinatorics.fibonacci.explore.ratioCallout
Fibonacci sequence
The Fibonacci sequence is defined by F_0 = 0, F_1 = 1, and F_n = F_{n-1} + F_{n-2} for n ≥ 2. It appears in nature (sunflower spirals, pinecones, nautilus shells) and in algorithms (Fibonacci heaps, dynamic programming).
Binet's closed-form: \(F_n = \frac{\varphi^n - (1-\varphi)^n}{\sqrt{5}}\)
\(F_n = F_{n-1} + F_{n-2}\)
Worked Examples
Golden ratio limit
\(\lim_{n\to\infty}\frac{F_n}{F_{n-1}} = \varphi \approx 1.618033\)