Question: 14. Prove by mathematical induction a. For n = 0, 1,2,3,4, .. ... p(n) : 1+5+5-+5'+..... +50- 5#1 1 4 i. Basic Step: ii. Inductive

14. Prove by mathematical induction a. For n = 0, 1,2,3,4, .. ... p(n) : 1+5+5-+5'+..... +50- 5#1 1 4 i. Basic Step: ii. Inductive Step: b. For n = 18, 19,20, ... P(n) : n is a sum of 7's and 4's. i. Basic Step: ii. Inductive Step: Show: For n = 18, 19,20,... p(n) - p(n + 1) Hint: Assume n 2 18 and p(m) is true (inductive hypothesis) Consider two cases: Casel: The sum for n has no 7, that is, it has only 4's. Then it must have at least five 4's because 1 > 18. etc 11 = 4+4+4+4+4+4k n+1 = 1 +4+4+4+4+4+4k=.... Case 2: The sum for n has a 7 n = 7+ 7k +4/ In+1 = 1+7+7k+4/ =.. ... etc

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