DISCRETE MATHEMATICS THROUGH GUIDED DISCOVERY ...

“It is possible to form postage totaling each of
3(N + 1) − 1, 3(N + 1), and 3(N + 1) + 1 cents
using only 3-cent and five-cent stamps.”
This is precisely the statement T (N + 1). Thus by the Principle of
Mathematical Induction, using only 3- and 5-cent stamps, you can make
n cents in postage for every n ≥ 8 .
The postage problem for any finite number of stamp types is also referred
to as Frobenius’ Problem and the Coin Exchange Problem.
Appendix B contains another review of the fundamentals of proofs using
the Principle of Mathematical Induction.
2.2 The Principle of Mathematical Induction
All proofs using the Principle of Mathematical Induction have four parts:
An identification of the sequence of statements to be proved, a base step,
an inductive step, and the inductive conclusion.
It is helpful to identify these four parts in the proof of the postage problem
in the last section. First of all, the sequence of statements S(n) which
must be proved was identified. In the postage problem this involved some
trial-and-error, whereas for (2.1) and (2.2) the statement of the problem
already identified the parametrized statement to be proved. For the postage
problem, the base step is the case n = 3. Next locate the sentence “Next
consider any integer N ≥ 3 for which T (N) is true, and then show that the
truth of T (N + 1) follows from the truth of T (N).” This is an outline of
what must be accomplished in the inductive step of the proof. In particular,
“Because T (N) is true, postage can be made for each of
3N − 1 , 3N , and 3N + 1”
is called the inductive hypothesis. In the inductive step the statement
is derived for n = N + 1 from the inductive hypothesis, proving that the
truth of the statement when n = N implies the truth of the statement
when n = N + 1. The last sentence in the proof, “Thus by the Principle of
Mathematical Induction, using only 3- and 5-cent stamps, you can make n
cents in postage for every n ≥ 8”, is the inductive conclusion.
One way of looking at the Principle of Mathematical Induction is that it
tells you that if you know the “first” case of a theorem and you can derive
every other case of the theorem from a smaller case, then the theorem is true
in all cases. However, the particular way in which this reasoning process has
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