Problem 2.[15 points] Complete this proof that n cents of postage can be formed using 3 and 5 cent stamps for all n 28 Proof. We use strong induction. (a)Let P(n)be the proposition that Solution. n cents of postage can be formed using 3 and 5 cent stamps (b) Base cases Solution. P(8), P(9), and P(10)are all true, since 9=3+3+3 (c) Inductive step Solution For n 10, we assume P( 8),..., P(n)and prove P(n+1). In particular, y assumption P(n-2), we can form n-2 cents of postage. Adding a 3-cent stamp gives n+1 cents of postage, so P(n+1) is true o P(n)is true for all n>8 by the principle of strong inductionQuiz 1 4 Problem 2. [15 points] Complete this proof that n cents of postage can be formed using 3 and 5 cent stamps for all n ≥ 8. Proof. We use strong induction. (a) Let P(n) be the proposition that Solution. n cents of postage can be formed using 3 and 5 cent stamps. (b) Base cases. Solution. P(8), P(9), and P(10) are all true, since: 8 = 5 + 3 9 = 3 + 3 + 3 10 = 5 + 5 (c) Inductive step. Solution. For n ≥ 10, we assume P(8), . . . , P(n) and prove P(n + 1). In particular, by assumption P(n − 2), we can form n − 2 cents of postage. Adding a 3-cent stamp gives n + 1 cents of postage, so P(n + 1) is true. So P(n) is true for all n ≥ 8 by the principle of strong induction