Is this proof for $sum^n_{i=1} i= frac{n^2+n}{2}$ correct?

Is this proof correct, as I feel unsure about whether or not I did that correct because the book did it differently, I wouldn't know however why my proof should be wrong.

Could you help me out?

The following statement is to be proven by induction. $$\sum^n_{i=1} = \frac{n^2+n}{2}$$ Base case $n=1$ $$1 = \frac{1+1}{2} \checkmark$$ Induction Step $n\rightarrow n+1$ $$\sum^{n+1}_{i=1}=\sum^n_{i=1}+(n+1)\\ \iff \frac{n^2+n}{2}+(n+1) \\ \iff \frac{n^2+n}{2}+\frac{2(n+1)}{2} \\ \iff \frac{n^2+n+2n+2}{2}\\ \iff \frac{n^2+3n+2}{2} \\ \iff \frac{(n+1)^2+(n+1)}{2}$$