MATH 137 : Calculus 1 for Honours Mathematics. Online Assignment #2. Introduction to Sequences

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1 1 MATH 137 : Calculus 1 for Honours Mathematics Online Assignment #2 Introduction to Sequences Due by 9:00 pm on WEDNESDAY, September 19, 2018 Instructions: Weight: 2% This assignment covers the topics from your Week 1 reading assignment in Section 1.2 Introduction to Sequences, Recursively Defined Sequences, Heron s Algorithm and Subsequences and Tails of Sequences from the Course Notes and the corresponding Chapter 1 online lectures. You will require this information to complete the following questions. Ensure you have also completed the Introduction to Sequences in Maple lab found in the Week 1 Tasks. Additionally, this assignment also includes a review of High School mathematics. Read and complete the following assignment. IMPORTANT: Keep a record of the answers you are going to submit on the printed copy of your assignment so that you can check your answers against the posted solutions. ***Completion of this assignment must be done by the student who submits it; additional help with this assignment is only permitted in accordance with the University of Waterloo s Academic Integrity rules. By submitting this assignment you understand and agree to the University s policies regarding academic honesty. Any violation of the University of Waterloo s Policy 71 Student Academic Discipline Policy (see on academic honesty will result in an academic penalty. To submit your solutions: Go to Waterloo LEARN course management site at learn.uwaterloo.ca Enter your QUEST Username and Password in the space provided and click Login. Click on the link for MATH 137 Online. Once inside the course website, click on the Syllabus icon, in the left-hand column click the Activities and Assignments link, and then click the Weekly Online Assignments link in the centre panel of the page. Read the guidelines and instructions about how to submit your assignments. When you are ready to submit your answers, click on the Submit tab near the top of the Math 137 home page in LEARN, and then click on Quizzes. Under the Submit Weekly Online Assignments section, click on Weekly Online Assignment 2. Read and follow any instructions provided. An answer key for this assignment will appear where you can fill-in your solutions. Please your instructor immediately if you encounter any problems (baforres@uwaterloo.ca). You should submit your solutions well in advance of the due date/time to avoid problems. Click on the SUBMIT button when you are done. You have only 1 attempt to submit your solutions. Any assignment submitted after 9:00 pm (Waterloo, Ontario time) on the due date will be considered LATE and will NOT be counted toward your final grade (no exceptions). Part 1: Multiple Choice: Choose the best answer. (1 mark each unless otherwise indicated) The following questions are a review of material from High School mathematics. If you have any difficulty with the following review questions, please contact your instructor for additional tutorial help. 1. Is the following statement true or false? is less than

2 2 2. Let b be any positive Real number. Is the following statement true or false? b 3 2 b 1 2 is equal to b 2 3. Let b be any positive Real number. Is the following statement true or false? b 3 is equal to b Is the following statement true or false? is equal to Solve: 1 x < a) x > 100 b) x > c) x (100, ) d) x (10000, ) e) Either (a) or (c) f) Either (b) or (d) 6. Let a 2 = 33 8, a 3 = and a 4 = Arrange these terms from smallest to largest. a) a 2, a 3, a 4 b) a 4, a 3, a 2 c) a 2, a 4, a 3 d) a 3, a 4, a 2 The following questions are based on material in your course notes and course lectures. 7. The expression x 3 < 2 can be read as: a) The distance between x and 3 is less than 2 units. b) The distance between 3 and x is less than 2 units. c) All of the above.. 8. If x < 0, then x is: a) x b) x c) 0 d) 1 e) None of the above 9. Is the following statement true or false? If {a n } is a sequence, then n is called the index of the sequence.

3 3 10. The formula for the general term a n of the sequence {1, 1 2, 1 4, 1 8,...} is: a) ( 1 2 )n b) 1 2n c) ( 1 2 ) n d) ( 1 2 )n 1 e) None of the above 11. The formula for the general term a n of the sequence {1, 1 2, 1 4, 1 8,...} is: a) 2 1 n b) 2 n 1 c) ( 2) n d) ( 2) n 1 e) ( 2) 1 n 12. What is the fourth term of the sequence { 3n+1 n+2 }? a) 1 2 b) 4 3 c) 7 4 d) 13 6 e) What is the fifth term of the sequence { cos(2nπ) n }? a) 1 2 b) 1 3 c) 1 4 d) 1 5 e) What is the fourth term of the recursive sequence {a n } where a 1 = 256 and a n+1 = a n? a) 256 b) 16 c) 4 d) 2 e) If {a n } is defined recursively by a 1 = 1 and a n+1 = a n + (n + 1), then a 3 =? a) 4 b) 6 c) 10

4 4 16. If {a n } is defined recursively by a 1 = 1 and a n+1 = a n + (n + 1), then a 45 =? (Hint: What is the formula for the sum of the first n integers?) a) 1035 b) 990 c) Not enough information to determine a If {a n } is defined recursively by a 1 = 1 2 and a n+1 = a n + ( 1 2 )n+1, then a 3 =? a) 3 4 b) 7 8 c) If {a n } is defined recursively by a 1 = 1 2 and a n+1 = a n + ( 1 2 )n+1, then which of the following statements is true? a) a n+1 > a n b) a n = 1 ( 1 2 )n c) a n < 1 d) All of the above 19. Let {F (n)} be the Fibonacci sequence which is given by F (1) = 1 = F (2) and F (n + 2) = F (n) + F (n + 1) for each n N. Recall that a natural number n is prime if n 2 and if n is only divisible by 1 and itself. How many of the first 12 terms of the Fibonacci sequence are primes? a) 4 b) 5 c) Let {F (n)} be the Fibonacci sequence. Which of the following statements is true? a) It is known the the sequence {F (n)} contains only finitely many prime numbers. b) It is known the the sequence {F (n)} contains infinitely many prime numbers. c) If a n = F (n+1) F (n), then a 1 = 1 and a n+1 = a n for each n. 21. Let {a n } be the sequence {1, 2, 3,..., n,...}. Which of the following statements is not true? a) The subsequence of {a n } consisting of all even natural numbers is given by {a 2k }. b) The subsequence of {a n } consisting of all odd natural numbers is given by {a 2k 1 }. c) The subsequence of {a n } consisting of all odd numbers is given by {a 2k+1 } Let {a n } be the sequence given recursively by a 1 = 1 and a n+1 = a n. Which of the following statements is true? a) a n < a n+1 for every n N. b) a n > a n+1 for every n N. c) a 2k < a 2k+2 and a 2k 1 > a 2k+1 for every k N. d) a 2k > a 2k+2 and a 2k 1 < a 2k+1 for every k N.

5 5 23. Given a sequence {a n }. Which of the following statements would not imply that a subsequence {a nk } contains a tail of {a n }? a) The subsequence {a nk } contains infinitely many terms in the original sequence {a n }. b) The subsequence {a nk } contains all but finitely many terms in the original sequence {a n }. c) The subsequence {a nk } is of the form {a N0+k} for some fixed N 0 N {0} Let {a n } be the sequence {1, 2, 3,..., n,...}. Which of the following subsequences is a tail of the original sequence {a n }? a) The subsequence {2, 4, 6,...} consisting of all even natural numbers. b) The subsequence of {1, 3, 5,...} consisting of all odd natural numbers. c) The subsequence of {14, 15, 16,...} consisting of the natural numbers greater than or equal to 14. d) All of the above. 25. Let {a n } be the sequence {( 1) n+1 } = {1, 1, 1, 1, 1, 1,...}. Which of the following intervals contains a tail of the sequence {a n }? a) ( 1 2, 1 2 ) b) ( 1 2, 3 2 ) c) Both a) and b). d) Neither a) nor b). 26. Let {a n } be any sequence. Which of the following pairs of intervals could both contain a tail of {a n } at the same time? a) I 1 = ( 1, 1) and I 2 = (1, 2) b) I 1 = [ 1, 1] and I 2 = [1, 2] c) Both a) and b). d) Neither a) nor b). The following questions are based on material from the Introduction to Sequences in Maple lab and the Recursive Sequences in Maple lab. It is assumed that the Maple CLASSIC version is used. Marking Scheme: The following Maple questions are worth 2 marks each. 27. Let a n = n2 (3 n 2 ). Create the plot of the first 20 terms of {a n} in Maple. Does the sequence seem to approach a particular value? Find the limit of {a n } in Maple. Did this limit seem to match the one indicated in the Maple plot of the sequence terms? What value do you think the terms in the tail of the sequence approach? a) 0 b) 1 c) 1 3 d) The terms never approach a particular value.

6 6 28. Let a n = cos(n)2 2 n. Create the plot of the first 20 terms of {a n } in Maple. Does the sequence seem to approach a particular value? Find the limit of {a n } in Maple. Did this limit seem to match the one indicated in the Maple plot of the sequence terms? What value do you think the terms in the tail of the sequence approach? a) 0 b) 1 c) {a n } oscillates so the terms do not approach one particular value. 29. Let a n = n! 2 n. Create the plot of the first 10 terms of {a n } in Maple. By studying the Maple plot, what value does it appear that the terms in the tail of the sequence of a n approach? a) 0 b) 1 2 c) 1 d) The terms do not seem to approach one particular value. END OF ASSIGNMENT: Submit your answers according to the instructions on page 1.

MATH 137 : Calculus 1 for Honours Mathematics Instructor: Barbara Forrest Self Check #1: Sequences and Convergence

1 MATH 137 : Calculus 1 for Honours Mathematics Instructor: Barbara Forrest Self Check #1: Sequences and Convergence Recommended Due Date: complete by the end of WEEK 3 Weight: none - self graded INSTRUCTIONS: