## is an homographic function.

Consider the equation .

1. If this equation has a double root , then define a sequence by . The sequence is arithmetic.
2. If this equation has a two distinct real solutions and , then define a sequence by . The sequence is geometric. The choice of the notations has no importance.

Example 3.9.2   Let be the sequence defined by its first term and the recurrence relation .

Let . We have:

This equation has two real solutions: 1 and . Define:

We have:

The sequence is geometric with ratio equal to , therefore it is converhent and has limit equal to 0. As , by Thm 7.1 the sequence is convergent and its limit is equal to 1.

Noah Dana-Picard 2007-12-28