Prove by induction that, for all :
Prove for :
LHS,
RHS,
True for , now assume for ,
Now the sequence above is for add numbers, assuming that is odd, then instead of using use .
We then have:
.
Not sure where to go from here? Any help would be great.
Thank you
It does not really matter if you noticed or not that it's the sum of cubes of odd numbers. This problem is solved in a completely general way. The problem asks to prove "For all , ", where is a property of ; in this case an equality that contains . The base case is to show , which you have already done. For the induction step, the induction hypothesis is . What you need to show is . It makes things much clearer if and are written explicitly. Note that is obtained from by replacing with .