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

Explanation

To prove the statement \( 1^3 + 2^3 + 3^3 + \ldots + n^3 = \left( \frac{n(n+1)}{2} \right)^2 \) for all \( n \in \mathbb{N} \) using the principle of mathematical induction, we follow these steps:

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

**Left Hand Side (LHS):**
\[
1^3 = 1
...