Answer
given statement is true for all integers n ≥ 14.
Work Step by Step
$Proof:$
Let P(n) be the property “n cents can be obtained by using 3-cent and 8-cent coins.”
-Showing that P(14) is true:
Fourteen cents can be obtained by using two 3-cent coins and one 8-cent coin.
-Showing that for all integers k ≥ 14, if P(k) is true, then P(k + 1) is true:
-Suppose k cents (where k ≥ 14) can be obtained using 3- cent and 8-cent coins.
[Inductive hypothesis]
- We must show that k + 1 cents can be obtained using 3-cent and 8-cent coins.
- If the k cents includes an 8-cent coin, replace it by three 3-cent coins to obtain a total of k + 1 cents.
- Otherwise the k cents consists of 3-cent coins exclusively, and so there must be least five 3-cent coins (since the total amount is at least 14 cents).
- In this case, replace five of the 3- cent coins by two 8-cent coins to obtain a total of k + 1 cents.
Thus,
- in either case, k + 1 cents can be obtained using 3-cent and 8-cent coins.
[This is what we needed to show.]
-[Since we have proved the basis step and the inductive step, we conclude that the given statement is true for all integers n ≥ 14.]
