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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.
- Summer '08