Prove by induction that if `n` is a positive integer not divisible by `3`, then `3^(2n)+3^(n)+1` is divisible by `13`.

Explanation

To prove that if \( n \) is a positive integer not divisible by \( 3 \), then \( 3^{2n} + 3^n + 1 \) is divisible by \( 13 \), we will use mathematical induction.

### Step 1: Base Case
Let's check the base case when \( n = 1 \):
\[ 3^{2 \cdot 1} + 3^1 + 1 = 3^2 + 3 + 1 = 9 + 3 + 1 = 13 \]
Since \( 13 \) is divisible by \( 13 \), the base case holds true.
...