GeoGebra

This is a 2-dimensional representation of the iteration rule $a_{n+2} = \frac{a_{n+1}+1}{a_n}$. The points plotted are of the form $\left(a_n,a_{n+1} \right)$. The two sliders control the values of the first two terms in the sequence. Drag either of the two sliders, and observe the motion of the individual points before you turn on the traces of the points. It is best to drag just one slider at a time.


Can you explain why the points move along the paths they do? Is it possible to find an equation that describes their paths? What are some invariants? Are there any characteristics of the traces that seem to persist regardless of the values of $a_1$ or $a_2$?

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