Fibonacci Numbers (Sequence) (2024)

$1,1,2,3,5,8,13,21,34,55,89,144,233,377,\mathrm{...}$

The Fibonacci numbers (The first $14$ are listed above) are a sequence of numbers defined recursively by the formula

$F_0=1$
$F_1=1$
$F_n=F_{n-2}+F_{n-1}$ where $n\ge 2$ .

Each term of the sequence , after the first two, is the sum of the two previous terms.

$1+1=2,1+2=3,2+3=5,3+5=8,5+8=13$ and so forth

This sequence of numbers was first created by Leonardo Fibonacci in $1202$ . It is a deceptively simple series with almost limitless applications. Mathematicians have been fascinated by it for almost $800$ years. Countless mathematicians have added pieces to the information regarding the sequence and how it works. It occurs throughout nature in things like patterns of spirals of leaves and seeds. It plays a significant role in art and architecture.

As you find the ratio of successive numbers in the Fibonacci sequence and divide each by the one before it, you discover that the value gets closer and closer to $\mathrm{1.61538...}$ , which is a close approximation of the Golden Ratio , whose exact value is $\frac{1+\sqrt{5}}{2}$ . The Golden Ratio is the ratio of the length to the width of the Golden Rectangle . Both of these are fascinating topics which warrant further research on your part.