## is an affine function.

There exists a real number such that the sequence determined by is geometric.

Example 3.9.1   Let be the sequence of real numbers defined by its first term and the recurrence relation . We look for a number such that the announced sequence will be geometric.

We have:

The sequence is geometric if, and only if, , i.e. .

Such a computation enables us to decide whether the sequence is convergent or not. Here the sequence is a geometric sequence whose ratio is equal to , thus it is convergent and its limit is 0. By Thm 7.1, we conclude that is convergent and its limit is equal to 12.

Noah Dana-Picard 2007-12-28