2345_2019-Spring-HW7-Induction-Recursion-Solutions.pdf - Math 2345 \u2013 Discrete Mathematics Homework 7 Induction Recursion Solutions Due Date

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Homework 7: Induction & Recursion Solutions Math 2345 – Discrete Mathematics Homework 7: Induction & Recursion Solutions Due Date: Monday, April 22, 2019 (beginning of class) Problem 7.1 (Integer Divisibility).For each integern0, letP(n)be the propositional function:n2+ 3nis even.(a) Prove thatP(0)is true.(b) WriteP(k).(c) WriteP(k+ 1).(d) Prove thatk0,[P(k)P(k+ 1)]is true.(e) Pull the results of (a)–(d) together and write an inductive proof thatn0, P(n)is true. (e) The solutions for (a) and (d) together show that the hypotheses of the Principle of Mathematical Inductionare true, therefore we can conclude that for every integern0, the integern2+ 3nis even.Problem 7.2 (Combinatorial Identities).Prove the following identity for every positive integern.1(1!) +· · ·+n(n!) = (n+ 1)!-1.
Homework 7: Induction & Recursion SolutionsProblem 7.3 (Coin Combinations).Use mathematical induction to prove that any amount of money of atleast14¢can be made up using a combination of3¢and8¢coins.Solution 7.3.14¢can be made with two3¢coins and one8¢coin.Therefore we have established a basis.Suppose thatkis a specific but arbitrarily-chosen integer wherek14and thatk¢can be made up using acombination of3¢and8¢coins (this is the inductive hypothesis). Consider(k+ 1)¢. Note to increase fromk¢to(k+ 1¢)we must add1¢but only use3¢and8¢coins. Consider two cases based on how many8¢coins areused to makek¢: