ws724 - Rob Bayer Instructions Math 55 MT2 Review Work...

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Rob Bayer Math 55 MT2 Review July 24, 2009 Instructions Work through the following review problems as a group Make sure to focus not just on getting the correct answers, but also on how you would actually write your proofs/solutions Feel free to skip around–there’s way more problems here than the actual midterm will have, so focus on whatever your group wants practice with. As always with review/practice tests, the inclusion or exclusion of certain topics should not be taken as an indication of what will be on the actual midterm. Induction and Recursion 1. Prove that 1 + 1 4 + 1 9 + · · · + 1 n 2 < 2 - 1 n for all positive n . 2. Consider the set of bit strings defined as followed: 1 S If w, v S , then w 0 S , and w 1 v S . 3. Use induction to prove that n 2 + 3 n is always even. (Can you think of a much easier proof?) Show that every string in S contains an odd number of 1 0 s 4. Find all solutions to the recurrence relation a n = - 2 a n - 1 + 15 a n - 2 + 4 n 5. Find a recurrence relation for the number of ways to tile a 1
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