Fibonacci Calculator — Find nth Term & Sequence

The Formula

The Fibonacci sequence is a series of numbers where each number (after the first two) is the sum of the two preceding ones. Defined by the recurrence relation F(n) = F(n-1) + F(n-2) with seed values F(0) = 0 and F(1) = 1, the sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144... The ratio of consecutive Fibonacci numbers converges to the golden ratio phi = (1 + sqrt(5))/2 ≈ 1.618, which appears throughout art, architecture, and nature.

Variable Definitions

How to Use This Calculator

1. 1

   Enter the position n (from 1 to 100) to find the nth Fibonacci number.

2. 2

   View the exact value of F(n) displayed prominently.

3. 3

   Scroll through the complete sequence of the first n terms, comma-separated.

4. 4

   Check the sum of the first n Fibonacci numbers for patterns and relationships.

5. 5

   Compare the iterative result with Binet's formula approximation, which becomes increasingly accurate for larger n.

Quick Reference

Common Applications

• Algorithm analysis: Fibonacci numbers measure the worst-case time complexity of Euclid's algorithm for GCD and appear in the analysis of Fibonacci heaps.
• Nature: Fibonacci numbers appear in plant phyllotaxis (leaf arrangement), pinecone spirals, sunflower seed heads, and nautilus shell growth patterns.
• Art and design: The golden ratio derived from Fibonacci numbers is used in composition, architecture (Parthenon), and visual design for aesthetic proportions.
• Financial markets: Some traders use Fibonacci retracement levels (23.6%, 38.2%, 61.8%) derived from the golden ratio for technical analysis.
• Computer science: Fibonacci numbers are used in data structures like Fibonacci heaps, in pseudorandom number generation, and as a test case for recursion and dynamic programming.

The Fibonacci spiral is constructed by placing squares with Fibonacci side lengths adjacent to each other and drawing a quarter-circle through each square.

Understanding the Concept

The Fibonacci sequence is one of the most famous and widely observed sequences in mathematics. Named after Leonardo of Pisa (Fibonacci), who introduced it in his 1202 book Liber Abaci, the sequence starts with F(0) = 0 and F(1) = 1, and each subsequent term is the sum of the two preceding ones. This seemingly simple recurrence generates numbers with remarkable properties. One of the most fascinating is the relationship to the golden ratio phi = (1 + sqrt(5))/2. As n grows large, the ratio of consecutive Fibonacci numbers F(n+1)/F(n) approaches phi. This connection is captured exactly by Binet's formula: F(n) = (phi^n - (-phi)^(-n)) / sqrt(5), which gives a closed-form expression for the nth Fibonacci number without iteration. For large n, the second term becomes negligible, giving the approximation F(n) = round(phi^n / sqrt(5)). The sum of the first n Fibonacci numbers has a simple closed form: sum of F(1) through F(n) = F(n+2) - 1. Fibonacci numbers also satisfy Cassini's identity: F(n-1) x F(n+1) - F(n)^2 = (-1)^n, which is useful for proving properties of the sequence. In nature, Fibonacci numbers appear in the spiral arrangements of leaves, the branching of trees, the fruitlets of pineapples, and the scales of pinecones, where they optimize packing efficiency. The sequence is also deeply connected to the Euclidean algorithm: the worst-case number of steps occurs when the inputs are consecutive Fibonacci numbers.

Frequently Asked Questions

Sources & References