Understanding the Fibonacci Numbers

The Fibonacci sequence begins with $0$ and $1$, and each subsequent term equals the sum of the two before it. Symbolically, if $F\left(n\right)$ represents the $n$-th term, then $F\left(n\right)=F(n-1)+F(n-2)$ for $n\ge 2$. This seemingly simple recurrence appears throughout mathematics and nature—from branching plants to the proportions of classical art.

Closed-form expressions connect these numbers to the golden ratio $\phi =\frac{1}{+}2$. Using Binet's formula, $F\left(n\right)=\frac{{\phi }^n-{(}^{-}}{\sqrt{5}}$, we see how the ratio between consecutive terms approaches $\phi$.

Using the Calculator

Enter a non-negative integer $n$ to compute the $n$-th Fibonacci number. The optional checkbox displays the entire sequence up to that term. Because the values grow quickly—approximately like $\phi n^{}$—the calculator limits $n$ to keep results readable. Experiment with different values to observe how the ratio $\frac{F}{(}F\left(n\right)$ approaches $\phi$.

Applications and History

The sequence appears in algorithms, financial models, and even population growth studies. Though named after Leonardo of Pisa—known as Fibonacci—similar sequences were studied in Indian mathematics centuries earlier. This tool lets you explore the numbers interactively and consider their remarkable properties.