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 andk° +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 - a1.) 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) → r2.) - p→ (q V r)) = p^ (¬q^¬r)

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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)

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter1: Fundamental Concepts Of Algebra

Section1.1: Real Numbers

Problem 36E

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