Answer: In order for a proof by mathematical induction to

Chapter 5, Problem 5.153

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In order for a proof by mathematical induction to be valid, the basis statement must be true for n =a and the argument of the inductive step must be correct for every integer k a. nd the mistakes in the proofs by mathematical induction. Theorem: For all integers n 1,3n 2 is even. Proof (by mathematical induction): Suppose the theorem is true for an integer k, wherek 1. That is, suppose that 3k 2 is even. We must show that 3k+1 2 is even. But 3k+1 2=3k32=3k(1+2)2 = (3k 2)+3k2. Now 3k 2 is even by inductive hypothesis and 3k2 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.