If $y_n=\frac{1-y_{n-1}}{y_{n-2}}$ and $y_1=a$ and $y_2=b$, find if the sequence repeats or not and for what value of $a, b$ does it repeats.
So I plug in $a=1$ and $b=2$ and here is the first few value for $y_n$:
$1,2,-1,1,0,1$ and $y_7$ is undefined.
But the difficult part is when I just use $y_1=a$ and $y_2=b$ and try to write the sequence: ( I will list the first few values for $y_n$)
$a,b, \frac{1-b}{a},\frac{a+b-1}{ab}, (1-a)a, \frac{(a^2-a+1)(ab)}{a+b-1}$...
I cannot really find a pattern here and if the sequence don't repeat, don't one have infinite pairs of $(a,b)$ that repeats? I am really confused here...
You really should have tried to find a pattern just a little bit harder. See: $$ \begin{array}{|l||c|c|c|c|c|c|c|c|}\hline \bf n & 1& 2& 3& 4& 5& 6& 7\\ \hline \bf y_n& a& b& 1-b\over a& a+b-1\over ab&1-a\over b\mathstrut& a& b\\ \hline \end{array} $$
In the process we have used cancelling out of the following terms: $a,b,1-a,\text{ and }1-b$. If any of these is 0, we'll run into an indeterminacy. Other than that, we'll be fine for any starting values.