Is $(-3)^n + 5^n$ monotone? How can I establish if this sequence is monotone?
If it isn't, is it permanently monotone from a certain n* to infinite?

Let $a_n=(-3)^n+5^n$. Then we have $a_n=2a_{n-1}+15a_{n-2}$. So as soon as you have two successive positive terms, it is easy to prove that the sequence is increasing.
Here we use that if $a_n=A\alpha^n+B\beta^n$ then $a_n=(\alpha+\beta)a_{n-1}-\alpha\beta a_{n-2}$ which comes from the general theory of recurrence relations.
