induction proof:
 
First let's see what Pk+1 would be:
[That's always the first thing to do.  Before you start an induction
proof,  you should calculate Pk+1 to see where you're headed]:
To do that, replace n by k+1 in  to see what Pk+1 is, 
for that is what we are going for, and if we have that beforehand,
we'll know when we have arrived and the proof is finished.

Substituting k+1 for n in , we have

 or 

 or



Now that we know what Pk+1 is, we know where we're going, 
and we'll know we have arrived if and when we get . 

So now we can start the proof:

P1:  substitute n=1, 
,
which is true.

Assume Pk: 

Add (k+1)³ to both sides:


 
Get an LCD of 4





Factor out (k+1)² in the numerator on the right:

 

Distribute on right:



Factor on right:





and now we see that we have gotten Pk+1 that we found in the 
beginning that we were going for.

So the proof is finished.

So since P1 is true, P1 proves P2, P2 proves P3, P3 proves P4,
etc., etc., ad infinitum.

Edwin