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?

To show that a sequence is stictly increasing, we must show that $a_{n+1} > a_n$ for all $n$.
\begin{array}
1(-3)^{n+1}+5^{n+1} &>& (-3)^n + 5^n \\
5^{n+1} - 5^n &>& (-3)^n - (-3)^{n+1} \\
5^n (5-1) &>& (-3)^n (1+3) \\
5^n &>& (-3)^n
\end{array}
When $n$ is even then we have $5^n > 3^n$ which is clearly true. When $n$ is odd we have $5^n > -3^n$ which is clearly true since $5^n$ is positive and $-3^n$ is negative. Hence your sequence increases for all $n\ge 1.$
