Continued Fraction for Root 5 How can I find the continued fraction expansion for the square root of 5. Do this without the use of a calculator and show all the steps.

We can use the 
$$x^2-5=0$$
$$x^2=5$$
$$x^2+x=5+x$$
$$x(x+1)=5+x$$
$$x=\frac{x+5}{x+1}$$
or
$$x=1+\frac{4}{1+x}$$
$$x=1\frac{4}{2\frac{4}{2\frac{4}{2\frac{4}{2.....}}}}$$
$$\sqrt{5}=1\frac{4}{2\frac{4}{2\frac{4}{2\frac{4}{2.....}}}}$$
