User:Egm6322.s12.team2.Xia/RP13.1

= R 13.1 - Continued fraction of $$\displaystyle \sqrt{5} $$ =

Given
$$\displaystyle \sqrt{5} $$

Find
Continued fraction

Solution
$$\displaystyle \sqrt{5}=2+(\sqrt{5}-2)=2+\frac{1}{\sqrt{5}+2}=2+\frac{1}{4+(\sqrt{5}-2)}=2+\frac{1}{4+\frac{1}{\sqrt{5}+2}}$$

$$\displaystyle =2+\frac{1}{4+\frac{1}{4+\frac{1}{4+\frac{1}{4+\frac{1}{\ddots}}}}} $$

thus

$$\displaystyle \sqrt{5}=2+\frac{1}{4+}\frac{1}{4+}\frac{1}{4+}\frac{1}{\ddots}=\{2:\overline{4}\}$$