Answer
See below.
Work Step by Step
1. Let $P(n)$ be the statement to be proved.
2. For $n=0$, we have $3^{0}-1=0$ and $0|8$, thus $P(0)$ is true.
3. Assume $P(k), k\gt0$ is true, that is $3^{2k}-1$ is divisible by 8.
4. For $n=k+1$, we have $3^{2k+2}-1=9\cdot3^{2k}-1=9(3^{2k}-1)+9-1=9(3^{2k}-1)+8$ which is divisible by 8.
5. Thus $P(k+1)$ is also true and we have proved the statement by mathematical induction.