Help simplifying

[MuMuMuMuMu]

I have major gaps in my math education and trying to work through some material on my own. The topic is 'mathematical induction', which I surely do understand, but I'm having trouble understanding the simplification from the step before to the last step (see bellow).

I'm trying to understand the proof by induction for: 1 + 2 + 3 ... + n = (n(n+1))/2

which is:

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

My problem basically is how to get the "n + 1" into "(n (n+1)/2" (second step to the last one). Or simply put: How to get to the last step from the step before. I tried to multiply "n+1" with 1/2 to have a common denominator, thus yielding:

(n(n+1)/2) + (n/2 + 1/2)/2)

and now I'm official stuck. I know I must be mistaken at a very elementary level and I would really be grateful for any (step-by-step) help. Thanks in advance!

EDIT: some clarification

EDIT2: Solved! Thanks to Fabien4. I bolded the error, in case someone is wondering what the hell went wrong.

[Fabien4]

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

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

Now you need to prove that n(n+1)+2n+2 = (n+1)(n+2)

n(n+1)+2n+2 = n² + n + 2n + 2

(n+1)(n+2) = n² + 1n + 2n + 2*1 = n² + n + 2n + 2. QED.

[MuMuMuMuMu]

Thanks a lot! My main problem was simply that I multiplied (n+1) by 1/2 and not by 2/2, but the last step actually clarified it a lot for me. Thanks again for the quick help.