Proper way to make transition from $k$ to $k+1$ in proofs by induction First off my apologies for asking really simple questions, but I'm having some trouble wrapping my head around proof of induction when trying to prove a statement involving a natural number n holds for all values of n. 
In the induction step lets assume the following simple example as found on this wikipedia page: 
Proof the formula below for all positive integers.

In the wikipedia example inductive step. They ADD k+1 on the left side, but REPLACE k with k+1 on the right side. My book adds k+1 on both the left AND the right side. Example: Wikipedia starts the right side of the inductive step with:

So the wikipedia example replaced n with k+1.
My book starts the right side of the inductive step with: $= \frac {k(k+1)}{2} + k+1$. So my book just added it to the right side. Now for this particular example it doesn't matter when you replace or add the k+1 on the right side. The outcome of the proof will be the same. 
My question is if it EVER matters in proof of induction. If I want to prove Pn+1 is true assuming Pn is true. Should I just replace the n with n+1 on both sides, should I add it on both sides of the equation or does it never matter?
EDIT: Based on multiple answers I was able to get a clear understanding of how I should properly use proof of induction! Thanks a lot everyone!

It does matter. However for this problem both methods amount to the same thing. 
In induction, you prove a base case and then prove that a next case will be true. Depending on your problem, this can take a range of approaches, as long as the approach proves the next case. 
