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I. Prove the following statements using the indicated proof. 1.) Let r €Z. If r is odd, then (r +3)2021 is even. (Direct Proof) 2.) Let a € Z. If (a – 1)6 +1 is odd, then a² – 2a +1 is even. (Contrapositive) 3.) Let r € Z. If r is odd, then (r +3)(r² + 7) is divisible by 32. (Direct Proof)

I. Prove the following statements using the indicated proof. 1.) Let r €Z. If r is odd, then (r +3)2021 is even. (Direct Proof) 2.) Let a € Z. If (a – 1)6 +1 is odd, then a² – 2a +1 is even. (Contrapositive) 3.) Let r € Z. If r is odd, then (r +3)(r² + 7) is divisible by 32. (Direct Proof)

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Definition 1: Let 1, y E Z. Then r is divisible by y if there exists an integer k such that r = ky. Definition 2: The product of two consecutive integers is always divisible by 2. For example, 2 + 3 and k° +4 are two consecutive integers for all integers k. Hence, its product (k + 3)(k² + 4) is divisibe by 2. I. Prove the following statements using the indicated proof. 1.) Let r € Z. If r is odd, then (r² + 3)2021 is even. (Direct Proof) 2.) Let a € Z. If (a – 1)6 + 1 is odd, then a? – 2a +1 is even. (Contrapositive) 3.) Let r € Z. If r is odd, then (r² + 3)(1² + 7) is divisible by 32. (Direct Proof) 4.) Let a € Z. If a is odd, then a? + (a + 2)² + (a + 4)² + 1 is divisible by 12. (Direct Proof) II. Prove the following statements using the Principle of Mathematical Induction (PMI). an+1 - a 1.) Let a +1 be a real number. Prove that a + a² + a³ + •..+a" for all integers n > 1. a - 2.) Let 1> -1 be a real number. Prove that (1+ 1)" 21 + nr for all integers n > 1. 3.) 2.7" +3- 5* – 5 is divisible by 24 for all integers n > 1. 4.) 10" + 3- 4*+2 + 5 is divisible by 9 for all integers n 2 1. III. Prove or disprove using truth table. 1.) (¬q V p) + r = (¬ p^ q) → r 2.) - p→ (q V r)) = p^ (¬q^¬r)
Transcribed Image Text:Definition 1: Let 1, y E Z. Then r is divisible by y if there exists an integer k such that r = ky. Definition 2: The product of two consecutive integers is always divisible by 2. For example, 2 + 3 and k° +4 are two consecutive integers for all integers k. Hence, its product (k + 3)(k² + 4) is divisibe by 2. I. Prove the following statements using the indicated proof. 1.) Let r € Z. If r is odd, then (r² + 3)2021 is even. (Direct Proof) 2.) Let a € Z. If (a – 1)6 + 1 is odd, then a? – 2a +1 is even. (Contrapositive) 3.) Let r € Z. If r is odd, then (r² + 3)(1² + 7) is divisible by 32. (Direct Proof) 4.) Let a € Z. If a is odd, then a? + (a + 2)² + (a + 4)² + 1 is divisible by 12. (Direct Proof) II. Prove the following statements using the Principle of Mathematical Induction (PMI). an+1 - a 1.) Let a +1 be a real number. Prove that a + a² + a³ + •..+a" for all integers n > 1. a - 2.) Let 1> -1 be a real number. Prove that (1+ 1)" 21 + nr for all integers n > 1. 3.) 2.7" +3- 5* – 5 is divisible by 24 for all integers n > 1. 4.) 10" + 3- 4*+2 + 5 is divisible by 9 for all integers n 2 1. III. Prove or disprove using truth table. 1.) (¬q V p) + r = (¬ p^ q) → r 2.) - p→ (q V r)) = p^ (¬q^¬r)
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