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

You assume that $P(n)$ is true and prove that $P(n+1)$ is also true from that.
So assume $1+2+3+\cdots+n=\frac {n(n+1)}2$.
Then from this we get $1+2+3+\cdots+n+(n+1)=\frac {n(n+1)}2 + (n+1)=(n+1)(\frac n2+1)=\frac {(n+1)(n+2)}2.$(Which the answer has avoided to mention here)
This is exactly what is done in that step you have stumbled upon. That's why writing the "$n$" matters in $1+2+3+\cdots+n+(n+1)$. Because you are using trueness of $P(n)$ here to show trueness of $P(n+1)$. 
