Can we prove that $\sum_{i=1}^n H_i=n^3?$

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A sequence of numbers \(H_1\), \(H_2\), \(H_3\), … is defined as follows: \(H_1=1\) and \(H_n=H_{n-1}+6(n-1)\) for \(n \ge 2\).

1. Write down the numerical values of \(H_2\), \(H_3\) and \(H_4\).

2. Find an expression for \(H_n\) in terms of \(n\).

3. Prove that \[\sum_{i=1}^n H_i=n^3.\]