hw3 - + F n-2 . (The rst few are 0, 1, 1, 2, 3, 5, 8, 13,...

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Math 55 - Homework 3 Due Wednesday, July 6, 2011 Directions: You are encouraged to discuss homework with fellow students. However, you should write up all solutions alone . Your answer should be clear enough that it explains to someone who does not already understand the answer why it works. (This is different than just convincing the grader that you understand.) 1. From section 4.1, do problem 70. 2. From section 4.1, do problem 76. 3. From Section 3.4, do problem 23. 4. In class, we prove that every number can be expressed as the sum of distinct powers of two. More generally, one can prove that every number is the sum of powers of b , where each power is used at most b - 1 times. (You don’t need to prove that, but it might be a good warm-up.) (a) Recall the Fibonacci numbers: F 0 = 0, F 1 = 1, and for n 2, F n = F n - 1
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Unformatted text preview: + F n-2 . (The rst few are 0, 1, 1, 2, 3, 5, 8, 13, 21.) Prove that every positive integer is the sum of distinct, nonconsecutive Fibonacci numbers. For example, 19 = 13 + 5 + 1. (b) Prove that every positive integer is the sum and dierence of distinct powers of three. For example, 76 = 81-9 + 3 + 1. 5. In class, we considered cutting a plane into regions with lines. Let R n be the maximum number of regions you can get with n lines. In class we showed that R n = n + R n-1 , and then used this to prove R n = 1 2 n 2 + 1 2 n + 1. Similarly, let S n be the maximum number of regions you can get by cutting space up using n planes. Find a relationship between S n and S n-1 , and use this to nd a formula for S n . Make sure to prove your answer....
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This note was uploaded on 03/14/2012 for the course MATH 55 taught by Professor Strain during the Summer '08 term at University of California, Berkeley.

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