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The second root is $-\omega^{-1}$. Every expression of form $A\omega^n+B(-\omega^{-1})^n$ is a solution of $X_{n+1}=X_{n}+X_{n-1}: further every solution is of this form for a solution is determined by $x_0$ and $x_1$, and values of $A$, $B$ can be found to satisfy $A+B=X_0$, $A\omega-B\omega^{-1}=x_1$, for any $x_0$, $x_1$. The number $\omega$ appears to have had a great power of fascination in the middle ages, due to the simple properties which follow from the equation which it satisfies. For instance a line divided in the ratio $\omega:1$ used to be described as divided in "golden section". As it will be convenient to have a word as well as a symbol for $\omega$ it will be described as the "golden ratio". The principle properties of this golden ratio, and its connections with the Fibonacci numbers, are summed up in the following.
Summary of the properties of Fibonacci numbers and the golden ratio
Let the Fibonacci numbers $F_n$ be defined by the condition
F_{n+1}=F_{n}+F_{n-1} (8.1)
F_0=0, F_1=1 (8.2)
and the golden ratio $\omega$ by the conditions
$\omega^2=\omega+1$ (8.3)
$\omega> 0$ (8.4)
Then
1) $F_{n+1}/F_n$ tends to the limit $\omega$ as $n$ tends to infinity.
2) The equations
$\omega^n=F_{n-1}+\omega F_n$ (8.5)
$\omega^{-n}=(-)^n (F_{n+1}-\omega F_n)$ (8.6)
$F_{-n}=(-)^{n-1} F_n$ (8.8)
$F_n=(1/\sqrt(5))(\omega^n-(-\omega^{-1})^{-n}) $ (8.8)
$F_n^2-F_{n-1}F_{n+1}=1$ (8.9)
hold for each $n$.