M122F07A3 - Math 122 F01 and F02 2007 Assignment 3 Due...

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Math 122 F01 and F02 2007 Assignment 3 Due : Friday, November 16, 2007, in class, before the lecture starts. Mathematical induction is a useful proof technique that every mathematician and computer scientist should know. The point of this assignment is to practice proofs that use this technique. Each induction proof must contain four parts: the Basis, the Induction Hypothesis (explicitly state what is assumed), the Induction Step, and the Conclusion. 1. a) [4] Let b 1 be an integer. Show that if n b and b n < n ! then b n +1 < ( n + 1)!. b) [3] What is needed to complete the proof that for any positive integer b we have b n < n ! for all large enough integers n ? Is this true for every positive integer b ? Justify your answer. (Hint: try looking at what you have when n b 3 .) c) [5] Prove that for all integers n > 7 we have 3 n < n !. 2. [8] Prove that any integer i 8 can be written as a sum of 3’s and 5’s. 3. [8] Notice that:
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