$f$ is an affine function.

There exists a real number $\alpha$ such that the sequence $(b_n)$ determined by $\forall n \in \mathbb{N}, b_n=a_n+ \alpha$ is geometric.

Example 3.9.1   Let $(a_n)$ be the sequence of real numbers defined by its first term $a_0=2$ and the recurrence relation $a_{n+1}= \frac 23 a_n +4$ . We look for a number $\alpha$ such that the announced sequence $(b_n)$ will be geometric.

We have:

$$b_{n+1} = a_{n+1} + \alpha$$

$$\quad = \frac 23 a_n + 4 + \alpha$$

$$\quad = \frac 23 (b_n - \alpha) +4 +\alpha$$

$$\quad = \frac 23 b_n + \frac 13 \alpha +4$$

The sequence $(b_n)$ is geometric if, and only if, $\frac 13 \alpha +4 =0$ , i.e. $\alpha = -12$ .

Such a computation enables us to decide whether the sequence $(a_n)$ is convergent or not. Here the sequence $(b_n)$ is a geometric sequence whose ratio is equal to $\frac 23$ , thus it is convergent and its limit is 0. By Thm 7.1, we conclude that $(a_n)$ is convergent and its limit is equal to 12.