what was i able to do
For any $n \in \mathbb Z$ worth $f (n + 1)− f (n) = a^{n+1}− a^n = a^n(a− 1)$, $a^n$ is always positive, so the sign of the difference depends on the sign of $a − 1$. If $a = 1$ is worth $f (n + 1) = f (n) \forall n \in \mathbb Z$ then $f$ is constant, if $a− 1 \lt 0$, $a \lt 1$ then $f (n + 1)− f (n) \lt 0$, $f (n + 1) \lt f (n)$, $f $ is decreasing and finally if $0 \lt a− 1$ , $a \gt 1$ then $f (n + 1) \gt f (n)$ and the function is increasing.
Mentioned properties are valid for all $n \in \mathbb Z $, for example in the case of $a \gt 1$ we have
· · · $\lt f (− 4) \lt f (− 3) \lt f (− 2) \lt f (− 1) \lt f (0) \lt f (1) \lt f (2) \lt f (3) \lt ·· · \lt f (n) \lt f (n + 1) \lt$ ·· ·
Thanks you for any help.