---
# System prepended metadata

title: Untitled

---

i)
$a_1:=1,a_{n+1}=1+\frac1{a_n},n\ge1$

$a_1 = 1$
$a_2 = 1+\frac11=2$
$a_3 = 1 + \frac 1 {2} = \frac 3 2 = 1,5$
$a_4 = 1 + \frac 1 {1,5} = 1 + \frac 2 3 = \frac 5 3 \approx 1,666$
$a_5 = 1 + \frac 1 {5/3} = 1 + \frac 3 5 = \frac 8 5 = 1,6$
$a_6 = 1 + \frac 1 {8/5} = 1 + \frac 5 8 = \frac{13}{8} = 1,625$
$a_7 = 1 + \frac 1 {13/8} = 1+ \frac 8 {13} = \frac {21}{13}$
$a_8 = 1 + \frac 1 {21/13}=1+ \frac {13} {21} = \frac {34} {21}$

$lim_{n\rightarrow\infty} a_n = \phi=\frac{1+\sqrt5}2\approx 1.61803$

ii)
$a_{n+2}<a_n,n$ gerade
$1+\frac1{1+{\frac1{a_n}}}<a_n$
$1+\frac1{\frac{a_n+1}{a_n}}<a_n$
$1+\frac{a_n}{a_n+1}<a_n$
$\frac{a_n+1+a_n}{a_n+1}<a_n$
$2a_n+1<a_n^2+a_n$
$0<a_n^2-a_n-1$
...
Gilt für $a_n>\phi$