Answer
See explanations.
Work Step by Step
Step 1. Prove the statement is true when $n=2$: $a_2=5\times3^{2-1}=15=3a_1$ , thus it is true for $n=1$.
Step 2. Assume the statement is true when $n=k$ ($k\gt2$): we have $a_k=5\times3^{k-1}$
Step 3. Prove it is true for $n=k+1$: we have $a_{k+1}=3a_k=3\times5\times3^{k-1}=5\times3^{k+1-1}$
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 $n$.