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ans0826

# ans0826 - Answers to in-class exercises Aug 26 1 Prove that...

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Answers to in-class exercises, Aug. 26 1. Prove that 2 is irrational. Proof. Suppose, on the contrary, that 2 were rational, that is, 2 = p / q for some integers p and q . We may assume without loss of gener- ality that p and q are in lowest terms (i.e., are relatively prime). Now 2 = p 2 q 2 , so p 2 = 2 q 2 . Therefore, p 2 is even. Since the product of two odd integers is odd (prove this), it must be the case that p is even, that is, p = 2 m for some integer m . Now 2 q 2 = p 2 = ( 2 m ) 2 = 4 m 2 , which implies q 2 = 2 m 2 . Hence q is even, so p and q are not in lowest terms, which contradicts the hypothesis. We conclude that 2 cannot be rational. 2. Use induction to prove that n i = 1 i 3 = n 2 ( n + 1 ) 2 4 . (1) Proof. Basis step: the result is clearly true for n = 1. Inductive step: Suppose that Eq. (1) holds for n . (This is not a vacuous assumption, since it’s true for n = 1.) We must show that Eq. (1) holds with n replaced by n + 1, that is, n + 1 i = 1 i 3 = ( n + 1 ) 2 ( n + 2 ) 2 4 .

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ans0826 - Answers to in-class exercises Aug 26 1 Prove that...

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