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 integern≥0, letP(n)be the propositional function:n2+ 3nis even.(a) Prove thatP(0)is true.(b) WriteP(k).(c) WriteP(k+ 1).(d) Prove that∀k≥0,[P(k)→P(k+ 1)]is true.(e) Pull the results of (a)–(d) together and write an inductive proof that∀n≥0, 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 integern≥0, 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 wherek≥14and 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¢:
