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 $\left(a_n,a_{n+1} \right)$.

Since point A has coordinates $\left(a_1,a_2 \right)$, moving point A will change both the two starting values. Click on point A and use the arrow keys on your keyboard to move the points to better control the motion of point A.

What happens as point A crosses the x-axis? What happens as point A crosses the y-axis? Do there appear to be any special points on the two axes? What happens as point A approaches the origin? What happens as point A moves away from the origin? Is it possible to have all five points occupy the same point at the same time? How would you characterize the motion of the five points?

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