Prove that: `1+2+3+ n<((2n+1)^2)/8` for all`""n in Ndot`

Explanation

To prove that \( 1 + 2 + 3 + \ldots + n < \frac{(2n + 1)^2}{8} \) for all \( n \in \mathbb{N} \), we will use the principle of mathematical induction.

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

\[ 1 < \frac{(2 \cdot 1 + 1)^2}{8} \]
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