Do you know how to do strong induction? For it, you suppose that a statement P holds. Then, if (P(0) holds and (whenever P(k) holds for all , then P(n+1) holds)), then P holds for all natural numbers.
In your case, we are amending this principle a little:
1. Instead of P(0), you have P(24)-P(28) (where P is the statement that the integer can be written as the sum of 5's and 7's).
2. In the assumption of the IH, instead of saying P(k) holds for all k<=n, we way it holds for all (i.e. we remove the first couple of numbers)
3. As we have shown P(24)-P(28), all we need to do is "start" the inductive hypothesis from P(29) on - i.e., assume n is greater or equal to 28, and that P(k) holds for all . We want to show that P(n+1) holds.
If we can do this, then this principle would show us that (the following is an intuitive discourse):
from P(28) holds: apply IH with n=28, and get that P(29) holds.
from P(29) holds: apply IH with n=29, and get that P(30) holds,... etc. Hence it holds for all natural numbers greater than 28, and as we've shown it holds for 24-28, it holds for all greater than 24.
(end of discourse)
So, how do we do (2)?
We have assumed and that (IH) the statement holds for all . How do we show that P(n+1) holds?
Consider n+1-5=n-4. As , what does our inductive hypothesis tell us about n-4? And what can we conclude about (n-4)+5=n+1?
This would conclude the induction, hence concluding the proof.
hey, i was trying to find an answer to this exact question, i have just started a course in discrete mathematics and are lecturers are pretty bad and don't really explain things well, we have been given this question in our first assignment and while i understand how to get the answer, i dont know how to properly set it out, i have never done strong induction before, so would you be able to write out the structure of teh answer to this quesiton, thanks
It would be a far more productive use of your time to actually sit down with your textbook and practice some of the examples (I suggest pp291-294). I somehow doubt that you'll be allowed forum access during the final exam. It would also be helpful to understand that you're studying a quite challenging topic; blaming the lecturers for your inability to understand is not going to help. If you're really having trouble, there are a variety of resources at your disposal. Speak to your tutors or visit the Nemeracy Centre.
Accept that you've been given a massive clue here, and move along. Begging for answers smacks of desperation.
PS, you've left your run a little late, haven't you? It's due in tomorrow.