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 $n\rightarrow n+1$ $$\sum^{n+1}_{i=1}=\sum^n_{i=1}+(n+1)\\ \iff \frac{n^2+n}{2}+(n+1) \\ \iff \frac{n^2+n}{2}+\frac{2(n+1)}{2} \\ \iff \frac{n^2+n+2n+2}{2}\\ \iff \frac{n^2+3n+2}{2} \\ \iff \frac{(n+1)^2+(n+1)}{2} $$
Yes it is correct indeed, also according to the usual way to write that foundamental identity, that is
$$\sum^n_{i=1} i= \frac{n(n+1)}{2}$$
at the end we obtain
$$\sum^{n+1}_{i=1} i= \frac{n^2+3n+2}{2}= \frac{(n+1)(n+2)}{2}$$
Refer also to the related