I need some help with mathematical induction, especially when related to Computer Science.

For example:

1+2+3..+n = n(n+1)/2

I have to proove for all n. Now the base case is n=1 so

1 = 1(1+1)/2 = 2/2 = 1

That part I get but now for any n+1 is where i get confused.

It is solved like this in a tutorial I read:

n+1 = ((n+1)((n+1)+1))/2 = ((n^2)+2n+1)/2

And thats the apparent answer but how does that proove it? I dont get it.

Thanks

Phil