b) Use mathematical induction to show that any quantity of at least 28 stamps can be obtained by buying a collection of 5-stamp packages and 8-stamp packages. Proof (by mathematical induction): Let the property P(n) be the following sentence. n stamps can be obtained by buying a collection of 5-stamp packages and 8-stamp packages. We will show that P(n) is true for every integer n 2 28. Show that P(28) is true: P(28) is true because 28 stamps can be obtained by buying 4 ✔8-stamp package(s). Show that for each integer k ≥ 28, if P(k) is true, then P(k+ 1) is true: Let k be any integer with k 2 28, and suppose that P(k) is true. P(k) is the sentence packages. [The supposition that P(k) is true is the inductive hypothesis.] We must show that P(k+ 1) is true. P(k+ 1) is the sentence ✔ 5-stamp package(s) and 1 stamps can be obtained by buying a collection of 5-stamp packages and 8-stamp packages. stamps can be obtained by buying a collection of 5-stamp packages and 8-stamp Assuming that P(k) is true, there are two possible cases: either the collection of packages that produces a total of k stamps includes three 5-stamp packages or it does not. Case 1 (k stamps are obtained from a collection that includes three 5-stamp packages): Let C be a collection of k stamps that includes three 5-stamp packages. Because C includes three 5-stamp packages, we can replace them with 2 ✔8-stamp packages to obtain one additional stamp, for a total of 16 stamps. X Case 2 (k stamps are obtained from a collection that does not include three 5-stamp packages): Let C be a collection of k stamps that does not include three 5-stamp packages. Then C includes at most 5-stamp packages, and so at most stamps in C come from 5-stamp packages. Thus, at least 8 x stamps come from 8-stamp packages, which implies that C contains at least 1 x 8-stamp packages because the smallest multiple of 8 that is greater than or equal to 18 is For a total of k + 1 stamps, replace k 8-stamp packages with k x 5-stamp packages to obtain one additional stamp. Therefore, in both of the two possible cases, k+ 1 stamps can be obtained by buying a collection of 5-stamp packages and 8-stamp packages. Hence, P(k + 1) is true [as was to be shown]. x