Now suppose \(P(k)\) is true for some arbitrary \(k \ge 28\text{.}\) Then it is possible to make \(k\) cents using 5-cent and 8-cent stamps. Note that since \(k \ge 28\text{,}\) it cannot be that we use fewer than three 5-cent stamps and fewer than three 8-cent stamps: using two of each would give only 26 cents. Now if we have made \(k\) cents using at least three 5-cent stamps, replace three 5-cent stamps by two 8-cent stamps. This replaces 15 cents of postage with 16 cents, moving from a total of \(k\) cents to \(k+1\) cents. Thus \(P(k+1)\) is true. On the other hand, if we have made \(k\) cents using at least three 8-cent stamps, then we can replace three 8-cent stamps with five 5-cent stamps, moving from 24 cents to 25 cents, giving a total of \(k+1\) cents of postage. So in this case as well \(P(k+1)\) is true.
in-context