Confused by Proof by induction 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?
$$P(n+1) = 1 + 2 + 3 + 4 + \cdots + (n+1) = \frac{(n+1)(n+2)}2$$

It's so that it is clear how to use the induction hypothesis. If the $n$ term wasn't visible, it wouldn't be so obvious how to proceed.
