assig1 - a n +1 and b n +1 are obtained from the terms a n...

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MATH 135 Algebra, Assignment 1 Due: Wed Sept 23, 8:30 am 1: (a) Find all real numbers x such that x 1 + x = 1 3 - x . (b) Solve y = x + 1 x with 0 < x 1 for x in terms of y . 2: (a) Find all ordered pairs of integers ( x,y ) such that xy = 6 + 2 x . (b) Find all ordered pairs of integers ( x,y ) such that x 2 + y 2 = 4 x + 2 y . 3: (a) Determine whether there exists an integer x such that x 2 + 20 is a perfect square. (b) Determine whether there exists an integer x such that x 2 + 10 is a perfect square. 4: (a) The first 15 odd prime numbers are 3 , 5 , 7 , 11 , 13 , 17 , 19 , 23 , 29 , 31 , 37 , 41 , 43 , 47 and 53. Determine which of these 15 prime numbers are equal to the sum of two squares, then make a conjecture about which odd prime numbers are equal to the sum of two squares. (b) Find all integers n with 2 n 20 with the property that n is a factor of 2 n - 1 - 1, then make a conjecture about which integers n 2 have the property that n is a factor of 2 n - 1 - 1. 5: (a) The first six terms of sequences { a n } and { b n } are listed below: n 1 2 3 4 5 6 a n 1 2 5 12 29 70 b n 1 3 7 17 41 99 (i) Find a rule that governs how the terms
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Unformatted text preview: a n +1 and b n +1 are obtained from the terms a n and b n , and use this rule to nd the terms a 7 , b 7 , a 8 and b 8 . (ii) Use a calculator to nd the ratio b n 2 /a n 2 for 1 n 6 and make a conjecture about the limit lim n ( b n /a n ). (b) For a real number x , let b x c denote the largest integer which is less than or equal to x . For each positive integer n , let a n = n 2 and let b n be the n th positive integer which does not occur in the sequence { a n } . The rst 20 terms of the sequence { a n } and the rst 8 terms of the sequence { b n } are listed below. n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 a n 1 2 4 5 7 8 9 11 12 14 15 16 18 19 21 22 24 25 26 28 b n 3 6 10 13 17 20 23 27 Make a conjecture about a rule which allows us to determine the terms a n and b n without using a calculator and without using the value of 2....
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This note was uploaded on 07/25/2011 for the course MATH 135 taught by Professor Andrewchilds during the Spring '08 term at Waterloo.

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