psol04 - Week 4 Homework Solutions Chapter 3 Section 1...

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Week 4 Homework Solutions Chapter 3 Section 1 12.The quadratic mean of a and b is always greater than or equal to the arithmatical mean of a and b. To prove we start by observing (a-b)2 S 0 and then by algebra that a 2 - 2ab + b 2 0 a 2 + b 2 2ab 2a 2 + 2b 2 a 2 + 2ab + b 2 (a 2 + b 2 )/2 [(a + b) 2 ]/4 sqrt ((a 2 + b 2 )/2) (a + b)/2 as desired. 20.We wish to show x 2 mod 5 = 1 or 4 when x mod 5 0. We consider four cases: Case 1: x mod 5 = 1. Then x 2 mod 5 = 1. Case 2: x mod 5 = 2. Then x 2 mod 5 = 4. Case 3: x mod 5 = 3. Then x 2 = 9 so x 2 mod 5 = 4 Case 4: x mod 5 = 4. Then x 2 = 16 so x 2 mod 5 = 1 In all cases, we have shown x 2 mod 5 = 1 or 4. 30.The product of primes of the form 4k+3 is a number of the form 4k+1: (4m + 3)(4n + 3) = 16mn + 12m + 12n + 9 = 4k + 1 where k = 4mn + 3m + 3n + 2 38.Let n be such that the sum of its divisors is n + 1. We note 1 and n are divisors of n and these sum to n + 1. Hence there are no other divisors of n so n is prime. 44.Let s = (a 1 )(a 2 ) . . . (a m ) and t = (b 1 )(b 2 ) . . . (b n ) be prime factorizations of s and t. Since gcf (s,t) = 1, a i b j for all i, j. The sum of the factors of s is the sum of all possible combinations of the a i and similarly for t and the b i . The product of these sums would then be the sum of all possible combinations of the a i and b j considered as a single group. Since the lists don't intersect, this is the same as the sum of the factors of st. Section 2: 6. A. 10, 7, 4, 1, -2, -5, -8, -11, -14, -17 B. 1, 3, 6, 10, 15, 21, 28, 36, 45, 55 C. 1, 5, 19, 65, 211, 665, 2059, 6305, 19171, 58025 D. 1, 1, 1, 2, 2, 2, 2, 2, 3, 3 E. 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 F. 1, 3, 7, 15, 31, 63, 127, 255, 511, 1023 G. 1, 2, 2, 4, 8, 11, 33, 37, 148, 153 H. 1, 2, 2, 2, 2, 3, 3, 3, 3, 3
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20. Σ n k=1 1/[k(k+1)] = Σ n k=1 (1/k - 1/(k+1)) = 1/k - 1/(n+1) 32.A. Countable. Let f(n) = (-1) n [ floor((n+1)/2) + floor(floor((n+1)/2)/3) ] B. Countable. Let f(n) = (-1) n [ 5 (floor((n+1)/2) + floor(floor((n+1)/2)/7))] C. Countable. Let g(n) = Σ n k=0
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This note was uploaded on 10/10/2009 for the course MATH 55 taught by Professor Strain during the Spring '08 term at University of California, Berkeley.

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psol04 - Week 4 Homework Solutions Chapter 3 Section 1...

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