Divisibility by induction

Base case. $n=1$: $3-1=2$ is even. ✓

Inductive step. Assume $3^n-1$ is even. Then
$$3^{n+1}-1=3\cdot3^n-1=3(3^n-1)+2.$$
Both terms are even (the first by induction, $2$ trivially), so the sum is even.

Conclusion. $3^n-1$ is divisible by $2$ for all $n\ge1$.