I have quite a few questions and so i just made it an image. Also attached.
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.
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
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!!
Originally Posted by moocav
...so by the inductive hypothesis the number can be written as a sum of no more than fractions with 1 as numerator, and thus...(!!!)