The terms <i>a</i><sub>1</sub>, <i>a</i><sub>2</sub>, <i>a</i><sub>3</sub>,...&#x00A0;<i>a</i><sub><i>n</i></sub>,...&#x00A0; of a
sequence are given by



<br clear="all" /><table border="0" width="100%"><tr><td>
<table align="center"><tr><td nowrap="nowrap" align="center">
<i>a</i><sub><i>n</i></sub> = </td><td nowrap="nowrap" align="center">
&#x00A0;1+<i>a</i><sub><i>n</i>&#8722;1</sub>
<div class="hrcomp"><hr noshade="noshade" size="1"/><i>a</i><sub><i>n</i>&#8722;2</sub><br /></td><td nowrap="nowrap" align="center">
.</td></tr></table>
</td></tr></table>


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.