GeoGebra

This is a number line representation of the $a_{n+2} = \frac{a_{n+1}+1}{a_n}$ iteration. The x-coordinates of the points A and B are $a_1$ and $a_2$, respectively. Points C, D, and E represent the next three terms in the sequence, in order. The semicircles provide another visual for tracking the orbits of the points. This visual is useful, for observing what happens when one terms in the sequence seem to run off to infinity.

Is it possible to have all the points at the same location on the number line? Point A represents the first term in the sequence. Is there always a number less than A in the sequence? In other words, is it possible for the first term to be less than every other term in the sequence?

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