Use mathematical induction (and the proof of Proposition as a model) to show that any amount of money of at least 14c can be made up using 3¢ and 8¢ coins.
Proposition
For all integers n ≥ 8, n¢ can be obtained using 3¢ and ¢ coins.
Proof (by mathematical induction):
Let the property P(n) be the sentence
n¢ can be obtained using 3¢ and 5¢coins. ← P(n)
Show that P(8) is true:
P(8) is true because 8¢can be obtained using one 3¢coin and one 5¢ coin.
Show that for all integers k≥ 8, if P(k) is true then P(k+1) is also true:
[Suppose that P(k) is true for a particular but arbitrarily chosen integer k ≥ 8. That is:]
Suppose that k is any integer with k ≥ 8 such that
k¢ can be obtained using 3¢ and 5¢ coins. ← P(k) inductive hypothesis
[We must show that P(k + 1) is true. That is:] We must show that
(k + 1)¢can be obtained using 3¢ and 5¢ coins. ← P(k + 1)
Case 1 (There is a 5¢ coin among those used to make up the k¢.): In this case replace the 5¢ coin by two 3¢ coins; the result will be (k + 1) ¢.
Case 2 (There is not a 5¢ coin among those used to make up the k¢.): In this case, because k ≥ 8, at least three 3¢ coins must have been used. So remove three 3¢ coins and replace them by two 5¢ coins; the result will be (k + 1) ¢.
Thus in either case (k + 1) ¢ can be obtained using 3¢ and 5¢ coins [as was to be shown].
[Since we have proved the basis step and the inductive step, we conclude that the proposition is true.]
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