Answer
See explanations.
Work Step by Step
Step 1. Prove the statement is true when $n=1$: $3^2-1=8$ is divisible by 8, thus it is true for $n=1$.
Step 2. Assume the statement is true when $n=k$: we have $3^{2k}-1$ is divisible by 8. We can assume $3^{2k}-1=8m$ or $3^{2k}=8m+1$ where $m$ is an integer.
Step 3. Prove it is true for $n=k+1$: $3^{2k+2}-1=9\times3^{2k}-1=9(8m+1)-1=72m+8=8(9m+1)$ which is divisible by 8.
Thus, the statement is also true for $n=k+1$
Step 4. Conclusion: with mathematical induction, we have proved that the statement is true for all natural numbers $n$.
