1. ## continued fraction approximations

Find the first 3 continued fraction approximations (as fractions not continued fractions) to:
1.(1+(squareroot 5))/2 (golden ratio)
2.(e-1)/2
Can anybody help with how to go about this as im not sure and the notes i have are not clear.
Thanks

2. Hello, studentsteve1202!

Find the first 3 continued fraction approximations to:

$\displaystyle 1)\;\;\frac{1+\sqrt{5}}{2}$ . (golden mean)

$\displaystyle 2)\;\;\frac{e^{-1}}{2}$

I know a very primitive method of doing this . . .

$\displaystyle \frac{1+\sqrt{5}}{2} \;=\;1.618033989\;=\;1 + 0.618033989 \;=\;1 + \frac{1}{\frac{1}{0.618033989}} \;=\;1 + \frac{1}{1.618033988}$

. . . . . $\displaystyle =\;1 + \frac{1}{1 + 0.618033988} \;=\; 1 + \frac{1}{1 + \frac{1}{1.61803988}} \;=\;1 + \frac{1}{1 + \frac{1}{1 + 0.618039988}}$

We see that the Golden Mean is:

. . $\displaystyle \phi \;=\; 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \hdots}}}$

The first three continued fractions are:

$\displaystyle a_1 \;=\;1$

$\displaystyle a_2 \;=\; 1 + \frac{1}{1} \;=\;2$

$\displaystyle a_3 \;=\;1 + \frac{1}{1 + \frac{1}{1}} \;=\;\frac{3}{2}$