How do you know when you're done with a proof by induction?

Question

So I'm doing a proof by induction, and in class the professor had some way of knowing when you were actually done with the proof. Before actually doing the proof he would write off to the side what his goal was to get the formula to look like and once he got it to look like that through factoring or whatnot then he knew he was done. But I don't know how he knows what that goal is and thus I don't know when I've actually finished the proof. This is the proposition followed by the work I've done so far.

For each natural number n, $1 + 5 + 9 + ... + (4n-3) = n(2n-1)$

Base Case:(n=1) $(4(1)-3)=1$ and $1(2(1)-1)=1$ so the formula holds for the base case

Assume the formula holds for n=k

$1+5+9+...+(4k-3)=k(2k-1)$

Show that the formula holds for k+1

$1+5+9+...+(4k-3)+(4(k+1)-3)=k(2k-1)+(4(k+1)-3)$

RHS = $k(2k-1)+4k+1$

$=2k^2 +3k+1$

$=(2k+1)(k+1)$

And this is as far as I've gotten with factoring the right side of the formula. And I can't really think of anything else I could do to it so I assume this is as far as I can go but I'm hoping someone can clarify to me the trick my professor did to know what I'm suppose to be working towards and therefore know when I'm done and have proved the proposition.

Answer 1

You are done. If you substitute $n=k+1$ in $n(2n-1)$ you get $(k+1)(2(k+1)-1)$ which is $(k+1)(2k+1)$. So you have completed the induction step.

Answer 2

To tell if the formula holds you must first know what the formula is. Nowhere in this post have you express what you are trying to demonstrate in the induction step. So writing and expressing what you are trying to do would be the first step.

In other words notice that your goal is to prove:

$1 + 5 + .... + (4k -3) + (4(k+1) -3) = (k+1)(2(k+1) - 1)$.

you have achieved showing:

$1 + 5 + .... + (4k -3) + (4(k+1) -3) = (2k+1)(k+1)$

Have you reached the goal? If not are you on the right track? What more need you do?

Well, since $(k+1)(2(k+1)-1) = (2k+1)(k+1)$, I would say, yes, yes you have reached your goal. (Or, at least, what more you need is minimum and obvious.)

Actually I'd have stated in the beginning that the goal is to show:

$1 + 5 + .... + (4k -3) + (4(k+1) -3) = (k+1)(2(k+1) - 1)= (k+1)(2k+1)$.

Then you would be completely done.

(That multiplication is commutative and $(k+1)(2k+1) = (2k+1)(k+1)$ is so very basic and obvious it utterly never needs to be stated or pointed out.)