Basic number theory / inequalities question

If p^3 + 1 = q^3 + 2q^2 and the conditions for p < q are to be determined, what is wrong with the following argument...

q > p implies that q^2 > p^2 and q^3 > p^3, so

p^3 + 1 > p^3 + 2p^2

?

Is it correct to say that the argument is describing a sufficient but not a necessary condition for p < q ?

What is the most obvious way to determine the conditions?

Thanks.

If p^3 + 1 = q^3 + 2q^2 and the conditions for p < q are to be determined, what is wrong with the following argument...

q > p implies that q^2 > p^2 and q^3 > p^3, so

p^3 + 1 > p^3 + 2p^2

?

Is it correct to say that the argument is describing a sufficient but not a necessary condition for p < q ?

What is the most obvious way to determine the conditions?

Really think you need to post the question. I think it's likely you've gone wrong before getting to your conditions, to be honest.

I have used my psychic abilities and determined the OP is trying to solve Step 1 2011 Q8.

I have used my psychic abilities and determined the OP is trying to solve Step 1 2011 Q8. Image below.



A note to the OP: note that the way this question is structured heavily hints towards repeating all the steps in (i) but for (ii). In other words, show that q<p<q+1 except for a certain range of q (using same methods in (a) and (b)), and investigate that range.

Thanks for the replies.


Haha those are impressive psychic abilities. I did the question as part of a mock exam and looking at it now I can't believe I chose to do it that way, it's ridiculous. Which makes your deduction all the more impressive!

By the way please could you tell me the name of the book you once recommended on here for reading before interviews? I think it was an oldish A-level textbook and I think I'm getting the authors name mixed up but I want to say it was by Martin Liebeck or someone or other.