{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

assig1

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

This preview shows page 1. Sign up to view the full content.

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 a n
This is the end of the preview. Sign up to access the rest of the document.

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 ﬁnd the terms a 7 , b 7 , a 8 and b 8 . (ii) Use a calculator to ﬁnd 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 ² and let b n be the n th positive integer which does not occur in the sequence { a n } . The ﬁrst 20 terms of the sequence { a n } and the ﬁrst 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....
View Full Document

{[ snackBarMessage ]}

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern