Im just confused by this equation and how it was used.

Assume

$$P(n) = 1 + 2 + 3 + 4 + \cdots + n = \frac{n(n+1)}2$$
Inductive step

$$P(n+1) = 1 + 2 + 3 + 4 + \cdots + n + (n+1) = \frac{(n+1)(n+2)}2$$

Why is the inductive step not written like this?

$$P(n+1) = 1 + 2 + 3 + 4 + \cdots + (n+1) = \frac{(n+1)(n+2)}2$$