A proof by mathematical induction is supposed to show that a given property is true for every integer greater than or equal to an initial value. In order for it to be valid, the property must be true for the initial value, and the argument in the inductive step must be correct for every integer greater than or equal to the initial value. Consider the following statement. For every integer n 2 1, 3" - 2 is even. The following is a proposed proof by mathematical induction for the statement. Since the property is true forn = 1, the basis step is true. Suppose the property is true for an integer k, where k 2 1. That is, suppose that 3k - 2 is even. We must show that 3k + 1 - 2 is even. Observe that 3k + 1 - 2 = 3k. 3 – 2 = 3k(1 + 2) – 2 = (3k – 2) + 3k . 2. Now 3K - 2 is even by inductive hypothesis, and 3K. 2 is even by inspection. Hence the sum of the two quantities is even by (Theorem 4.1.1). It follows that 3k +1- 2 is even, which is what we needed to show. Identify the error(s) in the proof. (Select all that apply.) The property is not true forn = 1. O 3k + 1- 2 + (3k – 2) + 3k . 2 O The inductive hypothesis is assumed to be true. V (3k - 2) + 3k . 2 * 3k(1 + 2) – 2 V 3k - 2 is odd by the inductive hypothesis.

A proof by mathematical induction is supposed to show that a given property is true for every integer greater than or equal to an initial value. In order for it to be valid, the property must be true for the initial value, and the argument in the inductive step must be correct for every integer greater than or equal to the initial value. Consider the following statement. For every integer n 2 1, 3" - 2 is even. The following is a proposed proof by mathematical induction for the statement. Since the property is true forn = 1, the basis step is true. Suppose the property is true for an integer k, where k 2 1. That is, suppose that 3k - 2 is even. We must show that 3k + 1 - 2 is even. Observe that 3k + 1 - 2 = 3k. 3 – 2 = 3k(1 + 2) – 2 = (3k – 2) + 3k . 2. Now 3K - 2 is even by inductive hypothesis, and 3K. 2 is even by inspection. Hence the sum of the two quantities is even by (Theorem 4.1.1). It follows that 3k +1- 2 is even, which is what we needed to show. Identify the error(s) in the proof. (Select all that apply.) The property is not true forn = 1. O 3k + 1- 2 + (3k – 2) + 3k . 2 O The inductive hypothesis is assumed to be true. V (3k - 2) + 3k . 2 * 3k(1 + 2) – 2 V 3k - 2 is odd by the inductive hypothesis.

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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