Use the principle of mathematical induction to show that `5^(2n+1)+3^(n+2).2^(n-1)` divisible by 19 for all natural numbers n.

Explanation

To prove that \( P(n) = 5^{2n+1} + 3^{n+2} \cdot 2^{n-1} \) is divisible by 19 for all natural numbers \( n \), we will use the principle of mathematical induction.

### Step 1: Base Case
We first check the base case \( n = 1 \).

\[ P(1) = 5^{2 \cdot 1 + 1} + 3^{1 + 2} \cdot 2^{1 - 1} \]
...