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Set 5.2 Questions - Exercise Set 5.2 1 U se mathematical...

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Exercise Set 5.2 1. Use mathematical induction (and the proof of Proposi- tion 5.2.1 as a model) to show that any amount of money of at least 14¢ can be made up using 3¢ and 8¢ coins. 2. Use mathematical induction to show that any postage of at least 12¢ can be obtained using 3¢ and 7¢ stamps. 3. For each positive integer n, let pen) be the formula 2 2 2 n(n + 1)(2n + I) 1+2+···+n= 6 . a. Write P(l). Is P(1) true? b. Write P(k). c. Write P(k + I). d. In a proof by mathematical induction that the formula holds for all integers n 2: I, what must be shown in the inductive step? 4. For each integer n with n 2: 2, let pen) be the formula 1/-1 " .. n(n-I)(n+l) 8 1 (1+1)= 3 . a. Write P(2). Is P(2) true? b. Write P(k). c. Write P(k + I). d. In a proof by mathematical induction that the formula holds for all integers n 2: 2, what must be shown in the inductive step? 5. Fill in Ole missing pieces in the following proof that I + 3 + 5 + ... + (2n - I) = n 2 for all integers n 2: I. Proof: Let the property pen) be the equation 2 I + 3 + 5 + ... + (2n - I) = n . ~ P(n) Show that P(l) is true: To establish P(l), we must show that when I is substituted in place of n, the left-hand side equals the right-hand side. But when n = I, the left-hand side is the sum of all the odd integers from I to I - I, which is the sum of the odd integers from I to I, which is just I. The right-hand side is ~, which also equals I. So PO) is true.
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