203w07hw5

# 203w07hw5 - EECS 203 Discrete Mathematics Winter 2007...

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EECS 203, Discrete Mathematics Winter 2007, University of Michigan, Ann Arbor Problem Set 5 Problems from the Textbook 9.1: 24[E] 9.2: 58[M] 9.3: 32[M], 68[M] 9.4: 16[E] 2.4: 26[C], 38[E] * Additional Problems Required for All Students Problem A5.1 (M) . Prove that in a party of an odd number of people, there must exist at least a person who has shaken hands with an even number of other people. Problem A5.2 (M) . The trace of a square matrix A R n × n is defined as trace( A ) def = n i =1 a ii . Let A, B R n × n be adjacency matrices for graphs G 1 and G 2 , respectively. Prove that if G 1 and G 2 are isomorphic, trace( A r ) = trace( B r ) for any integer r 1. Problem A5.3 (C) . Let n 0 be an integer, x R , and | x | < 1. Give, and prove the correctness of, a close form expression (that is, a simple function that does not involve infinite sums) for k =1 k 2 x k . * Extra Credit
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Unformatted text preview: Problem A5.4 ( Extra Credit). In this problem you will be asked to calculate the monthly payment of a mortgage. Suppose you borrow P dollars for M months at a rate r , starting in the ﬁrst day of Month 0. At the end of each month i , i = 0 , 1 ,...,M-1, you pay q i , the interest for that month, which is P i · ( r/ 12), where P i is the balance at the beginning of Month i . You also pay p i dollars toward the principle, so that the balance of the principle becomes P i +1 = P i-p i for the next month. In real life, your lender will “amortize” the payment so that p i + q i remains constant for all i , and you pay oﬀ your loan at the end of Month M-1. Question: give close form expressions for p i and q i . 1...
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• Winter '07
• YaoyunShi
• University of Michigan

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