DISCRETE MATHEMATICS THROUGH GUIDED DISCOVERY ...

Here
S(n) is the statement
n�
i=1
1 n
=
i · (i + 1) n + 1 .
Then observe that the left-hand sum in statement S(N + 1) is obtained by
adding 1/(N +1)·(N +2) to the left-hand sum in the immediately preceding
statement S(N). Because of this, the inductive process is basically the same
for (2.1) and (2.2), although the algebra involved is somewhat different.
•49. Prove each of (2.1) and (2.2) by induction on n ≥ 1.
•50. What postage do you think can be made using only three-cent and
five-cent stamps? If you have an unlimited supply of these two types
of stamps, do you think that there is a number N such that for every
n ≥ N, you can make n cents worth of postage?
In the last problem you probably did some calculations to convince yourself
that n cents worth of postage can be made for n = 3, 5, 6 (but not for
n = 1, 2, 4, 7) and also that apparently postage can be made for all integers
that are at least 8. A proof of this fact will now be given in full detail.
First of all, the sequence of statements to be proved is:
S(n): n cents of postage can be made using only 3− and 5−cent
stamps.
A key observation (and one that highlights how this problem is different
from proving the identities (2.1) and (2.2) ) is that you can not prove S(13)
is true by working only with the fact that S(12) is true. The reason for this
is simple: you do not have a 1-cent stamp at your disposal. In general you
can never conclude that n cents of postage can be made simply by knowing
that n − 1 cents of postage can be made; that is, in this example the truth
of the statement S(n) cannot be proved using only the truth of S(n − 1).
As is done in any proof by induction, there will first be an analysis of
some small cases. You’ve already noted in the last problem that the base
integer must be b ≥ 8; this is forced by the fact that 7 cents of postage cannot
be made using only 3-cent and 5-cent stamps. Also, you’ve undoubtedly
noted that you can make 8 cents with one 3-cent stamp and one 5-cent
stamp; 9 cents can be made with three 3-cents and 10 with two 5-cents.
Maybe you’ve checked larger integers as well, but at this point the truth of
each of S(8), S(9), S(10) has been verified.
It has already been observed that you can not get 13 cents from the fact
you can get 12 cents. One way to get 13 cents is to add a 3-cent stamp to
the two fives used to get 10-cents. The surprising thing is that this simple
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