By the principle of mathematical induction prove that for every `n in N, ` the following statements are true:
`1 +5 +9 + ....+ (4n - 3) = 2n^2 - n`

Explanation

To prove the statement \(1 + 5 + 9 + \ldots + (4n - 3) = 2n^2 - n\) for every \(n \in \mathbb{N}\) using the principle of mathematical induction, we will follow these steps:

### Step 1: Base Case
We start by checking the base case for \(n = 1\).

**Left-Hand Side (LHS):**
\[
1 + 5 + 9 + \ldots + (4 \cdot 1 - 3) = 1
...