Answer
$a_n=4$
See explanations.
Work Step by Step
Step 1. Find an explicit formula: $a_1=4, a_2=3a_1-8=4, a_3=3a_2-8=4, ... $, thus we get $a_n=4$
Step 2. Prove the formula is true when $n=1$: $a_1=4$ , thus it is true for $n=1$.
Step 3. Assume the formula is true when $n=k$: we have $a_k=4$
Step 4. Prove it is true for $n=k+1$: $a_{k+1}=3a_k-8=3\times4-8=4$
Thus, the formula is also true for $n=k+1$
Step 5. Conclusion: with mathematical induction, we have proved that the formula is true for all natural numbers $n$.
