Can someone explain to me this :
Why we use both, "n" and "n+1" in the third stage if math induction (where we check if statement holds for "n+1". I'll give an example
Prove that $1+2+3+...+n=\frac{n(n+1)}{2}$
When I get to the third step, I should write : $1+2+3+...+n+(n+1)=$ doesn't matter My question is, why I can't write $1+2+3+...+(n+1)$? If I must write $n$ in front of $n+1$, shouldn't I write $(n-1)$ where it goes to $n$ ? I hope that you will understand my question.