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Unformatted text preview: MATHEMATICS 61 SPRING 2010 First Midterm Examination April 19, 2010 Name: Signature: UCLA ID Number: Instructions: No calculators, books, or notes are allowed. Answer all 6 questions. Use only the scratch paper provided. 1 1 (18 points). Prove that, for every positive integer n , 2 3 n n i =1 (3 i 1) n i =1 3 i . Solution. We use induction. Basis Step: 1 i =1 (3 i 1) 1 i =1 3 i = 3 1 1 3 1 = 2 3 = 2 3 1 . Inductive Step: Assume that the statement is true for n . n i =1 (3 i 1) n i =1 3 i = n i =1 (3 i 1) n i =1 3 i 3 n + 2 3( n + 1) 2 3 n 3 n + 2 3( n + 1) = 3 n + 2 3 n 2 3( n + 1) 2 3( n + 1) . The first is by the induction hypothesis. 2 2 (17 points). Find a positive integer k such that: (a) postage of k cents or more can be achieved by using only 3 and 8 cent stamps; (b) k is the least number with property (a). Prove that your k satisfies both (a) and (b). Solution. To find the least k that works, we look for three consecutive numbers that are of the form 3 a + 8 b for nonnegative integers a and b . We find that the first such numbers are 14, 15, and 16. It is easy to see that 13 is not of the form, and 14 = 3 2 + 8; 15 = 3 5; 16 = 8 2 . Let k = 14. To show that 14 satisfies (a), we use the strong form of induction. Let n 14. Assume that every number m with 14 < n is of the form 3 a +8 b . We have already proved that n has this form if n is 14, 15, or 16. If n > 16, then n 3 14, and so the induction hypothesis implies that n 3 = 3 a +8 b for some a and b . Thu n = 3( a + 1) + 8 b . Because 13 is not of the required form, no number < 14 satisfies (a), so 14 satisfies (b). 3 3 (16 points). Let Z + be the set of all positive integers. Let R be the relation on Z + Z + defined as follows: ( m,n ) R ( m ,n ) (1) max( m,n ) < max( m ,n ) or (2) max( m,n ) = max( m ,n ) & m < m or (3) max( m,n ) = max( m ,n ) & m = m & n n ....
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This note was uploaded on 10/16/2010 for the course MATH 61 taught by Professor Enderson during the Spring '08 term at UCLA.
 Spring '08
 Enderson

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