Theorem: If a and b are consecutive integers, then the Choose a match sum a + b is odd. Proof: Assume that a and b are consecutive integers. Because Proof by Contradiction a and b are consecutive we know that b = a + 1. Thus, the sum a +b may be re-written as 2a + 1. Thus, there exists a number k such that a + b = 2k + 1 so the sum a + Proof by Contrapositive b is odd. Direct Proof Theorem: If a and b are consecutive integers, then the sum a + b is odd. Proof: Assume that a and b are consecutive integers. Assume also that the sum a + b is not odd. Because the sum a + b is not odd, there exists no number k such that a + b = 2k + 1. However, the integers a and b are consecutive, so we may write the sum a +b as 2a + 1. Thus, we have derived that a+ b k+1 for any integer k and also that a + b = 20 + 1. If we hold that a and b are consecutive then we know that the sum a +b must be odd.

Theorem: If a and b are consecutive integers, then the Choose a match sum a + b is odd. Proof: Assume that a and b are consecutive integers. Because Proof by Contradiction a and b are consecutive we know that b = a + 1. Thus, the sum a +b may be re-written as 2a + 1. Thus, there exists a number k such that a + b = 2k + 1 so the sum a + Proof by Contrapositive b is odd. Direct Proof Theorem: If a and b are consecutive integers, then the sum a + b is odd. Proof: Assume that a and b are consecutive integers. Assume also that the sum a + b is not odd. Because the sum a + b is not odd, there exists no number k such that a + b = 2k + 1. However, the integers a and b are consecutive, so we may write the sum a +b as 2a + 1. Thus, we have derived that a+ b k+1 for any integer k and also that a + b = 20 + 1. If we hold that a and b are consecutive then we know that the sum a +b must be odd.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.2: Mathematical Induction

Problem 50E: Show that if the statement 1+2+3+...+n=n(n+1)2+2 is assumed to be true for n=k, the same equation...

See similar textbooks