Fibonacci is probably the most well known sequence in mathematics. Today, we will see how Fibonacci numbers can be used to construct beautiful patterns called

*continued fractions*.

Fibonacci numbers are the following $$0, ~1, ~1, ~2, ~3, ~5, ~8, ~13, ~21, ~34, ~55, ~89, ~144, \dots$$ We can easily see the pattern of this number sequence. Each Fibonacci number is equal to the sum of the previous two numbers.

Mathematically, we can write the formula as follows $$F_0 = 0, ~F_1 = 1, ~~F_n = F_{n-1} + F_{n-2}$$

Now, let us consider the Fibonacci fractions $\frac{F_n}{F_{n-1}}$.

First, $$\frac{F_2}{F_1} = \frac{1}{1}$$

Next, $$\frac{F_3}{F_2} = \frac{2}{1} = 1 + \frac{1}{1}$$

$$\frac{F_4}{F_3} = \frac{3}{2} = 1 + \frac{1}{2} = 1 + \cfrac{1}{\frac{2}{1}} = 1 + \cfrac{1}{1 + \cfrac{1}{1}}$$

$$\frac{F_5}{F_4} = \frac{5}{3} = 1 + \frac{2}{3} = 1 + \cfrac{1}{\frac{3}{2}} = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1}}}$$

That is so cool, isn't it?!

$$\frac{F_6}{F_5} = \frac{8}{5} = 1 + \frac{3}{5} = 1 + \cfrac{1}{\frac{5}{3}} = 1 + \cfrac{1}{\frac{F_5}{F_4}} = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1}}}}$$

So we can follow the pattern and derive the following beautiful continued fractions

$$\frac{F_7}{F_6} = \frac{13}{8} = 1 + \cfrac{1}{\frac{F_6}{F_5}} = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1}}}}}$$

$$\frac{F_8}{F_7} = \frac{21}{13} = 1 + \cfrac{1}{\frac{F_7}{F_6}} = 1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1 + \cfrac{1}{1}}}}}}$$

Let us stop here for now. You can read more about Fibonacci numbers here. Hope to see you again next time.