To prove this formula by induction on $n$, we need to follow these steps:

Statement of the Formula

The formula we are proving is: $(n+1) \sum_{i=1}^n i^k = \sum_{i=1}^n i^{k+1} + \sum_{p=1}^n \left( \sum_{i=1}^p i^k \right).$

Step 1: Base Case $n = 1$

For $n = 1$, we substitute $n = 1$ in both sides of the equation and check if they are equal.

Left-hand side:

$(1+1) \sum_{i=1}^1 i^k = 2 \cdot 1^k = 2.$

Right-hand side:

[ \sum_{i=1}^1 i^{k+1} + \sum_{p=1}^1 \left( \sum_{i=1}^p i^k \right)