Prove the following by using the principle of mathematical induction for all `n in N`:`1. 3 + 3. 5 + 5. 7 + ... + (2n 1) (2n + 1)=(n(4n^2+6n-1))/3`

Explanation

To prove the statement \( P(n): 3 + 3 \cdot 5 + 5 \cdot 7 + \ldots + (2n - 1)(2n + 1) = \frac{n(4n^2 + 6n - 1)}{3} \) for all \( n \in \mathbb{N} \) using the principle of mathematical induction, we will follow these steps:

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

**Left Hand Side (LHS):**
\[
LHS = (2 \cdot 1 - 1)(2 \cdot 1 + 1) = 1 \cdot 3 = 3
...