I have quite a few questions and so i just made it an image. Also attached.

Induction Questions
Only need help with questions

**2** and

**5**
Oh and so far my lecturer has taught well-ordering, strong induction and simple induction. But I could only follow simple induction... the other two I'm quite clueless about >< Though tell me which method is best for each question.

**Attempts** __Question 2__
I have no clue to how to start it..

all i've done is

24 = 7 + 7 + 5 + 5

25 = 5 + 5 + 5 + 5 + 5

26 = 7 + 7 + 7 + 5

27 = 7 + 5 + 5 + 5 + 5

28 = 7 + 7 + 7 + 7

29 = 7 + 7 + 5 + 5 + 5

no idea what to do next

This was already practically solved some weeks ago...(Angry)...**sigh**...take any natural number $\displaystyle n>29\Longrightarrow \exists k\in\mathbb{N}\,\,s.t.\,\,\,24\leq n-5k\leq 29$ (why? Think!) , so $\displaystyle n-5k$ can be written as a sum of 5's and 7's...and voila! *Question 5*
Show that n/t - 1/(q+1) is positive and numerator is less than n

where t = nq + r with 0 < r < n

(get common denominator then expand and simplify)

n/t - 1/(q+1)

= n(q + 1)/[t(q+1)] - t/[t(q+1)]

= [n(q+1) - t] / [t(q+1)]

= [nq - t + n] / [t(q+1)]

t = nq + r

nq - t = -r

hence n/t - 1/(q+1)

= [n-r] / [t(q+1)]

from 0 < r < n

n > r therefore n - r > 0 (proved that numerator is positive)

and since r > 0 then n - r < n (proved that numerator is < n)

I'm not sure where to go from here

*Please someone help me however you can..* *Thank you in advance!!*