Fibonacci Calculator
Calculate the nth Fibonacci number using the recurrence F(n) = F(n-1) + F(n-2), with options for custom starting values.
Position in the Fibonacci sequence (n ≥ 0)
Results
F(n)
55
F(n)/F(n−1) (→ φ ≈ 1.618)
1.6176470588
How to Use This Calculator
Enter your input values
Fill in all required input fields for the Fibonacci Calculator. Follow the units and input format printed next to each field. Where a unit selector is available, choose the units that match your data.
Review your inputs
Double-check that all values are correct and match the units or format shown for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.
Read the results
The Fibonacci Calculator computes the output in your browser and displays the results, with units clearly labeled, as soon as they are ready — no data sent to a server.
Explore parameter sensitivity
Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.
When to Use This Calculator
- •Use the Fibonacci Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
- •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
- •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
- •Use it to explore the behavior of mathematical functions across a range of inputs.
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About Fibonacci Calculator
The Fibonacci sequence is arguably the most famous sequence in mathematics. Beginning with 0 and 1, each subsequent number is the sum of the two preceding ones: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. Named after Leonardo of Pisa (known as Fibonacci), who introduced the sequence to Western mathematics in his 1202 book Liber Abaci, the sequence models rabbit population growth in his original problem. The Fibonacci sequence appears throughout nature — in the spirals of sunflower seeds, the branching of trees, the arrangement of pine cone scales, and the shell of the nautilus. The ratio of consecutive Fibonacci numbers converges to the golden ratio φ ≈ 1.6180339887, a value prized in art, architecture, and design for its aesthetic properties. In computer science, Fibonacci numbers arise in the analysis of algorithms (Fibonacci heaps, Euclid's algorithm complexity), data structures, and dynamic programming exercises. This calculator computes F(n) iteratively for accuracy with large n and also shows how the ratio of consecutive terms approaches φ.
The Math Behind It
Formula Reference
Recurrence
F(n) = F(n−1) + F(n−2)
Variables: F(0) = 0, F(1) = 1 (standard)
Binet's Formula
F(n) = (φⁿ − ψⁿ) / √5
Variables: φ = (1+√5)/2 ≈ 1.618, ψ = (1−√5)/2 ≈ −0.618
Worked Examples
Example 1: Compute F(10)
Find the 10th Fibonacci number (0-indexed).
F(10) = 55
Example 2: Golden Ratio Approximation
Show that F(10)/F(9) approximates φ.
The ratio 55/34 ≈ 1.6176 is within 0.025% of φ.
Common Mistakes & Tips
- !Off-by-one errors: F(0) = 0, F(1) = 1 in 0-indexed convention; some sources use 1-indexed.
- !Using Binet's formula for large n without arbitrary-precision arithmetic — floating-point errors accumulate.
- !Assuming the Fibonacci sequence always starts at 0 and 1; generalized sequences can start with any two values.
Related Concepts
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Frequently Asked Questions
What is the golden ratio and how does it relate to Fibonacci?
The golden ratio φ = (1 + √5)/2 ≈ 1.618 is the limit of the ratio of consecutive Fibonacci numbers. As n grows, F(n+1)/F(n) approaches φ with increasing accuracy.
How fast do Fibonacci numbers grow?
Fibonacci numbers grow exponentially, approximately as φⁿ/√5. F(n) has roughly n × log₁₀(φ) ≈ 0.209n digits.
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