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Unformatted text preview: Rob Bayer Math 55 Worksheet July 10, 2009 Induction 1. Prove that 1 + 2 3 + 3 3 + 4 3 + + n 3 = n ( n + 1) 2 2 2. Find the flaw in each of the following proofs. (a) Claim: In any group of n people, all n have the same birthday. Proof: Well go by induction on n . If n = 1, then clearly that person has the same birthday as themself. Suppose the claim holds for groups of k people. Well show it holds also for groups of k + 1. Suppose we have some group of k + 1 people, numbered 1 through k + 1. By the IH, 1 through k all have the same birthday and 2 through k + 1 all do too. Since there is someone shared between these two groups, all k + 1 must share the same birthday. Thus, by induction, the result holds for all n (b) Claim: All integers are perfect squares. Proof: Clearly 1 is a perfect square. Suppose the claim works for integers up to and including k . Then if we write k + 1 = ab , the IH tells us that a = m 2 and b = l 2 for some integers m,l . Thus, k + 1 = m 2 l 2...
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This note was uploaded on 09/10/2009 for the course MATH 55 taught by Professor Strain during the Summer '08 term at University of California, Berkeley.
- Summer '08