Challenge Level

The terms $a_1, a_2, a_3,...\ a_n,...\ $ of a sequence are given
by: $$a_n =\frac{1+a_{n-1}}{a_{n-2}}.$$ Investigate the sequences
you get when you choose your own first two terms. Make a conjecture
about the behaviour of these sequences. Can you prove your
conjecture? Investigate the sequences.