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MATA37 2011summer

# MATA37 2011summer - ,a 3 as follows a 1 = 3 a 2 = 6 a n = 5...

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University of Toronto at Scarborough MAT A37H Summer 2011 Assignment # 1 You are expected to work on this assignment prior to your tutorial during the week of May 16th. You may ask questions about this assignment in that tutorial. At the beginning of your LECTURE on Thursday May 26th you need to submit the following homework problems. Note that this is not our typical submission procedure. We have this alteration in submission procedure for this week only because Monday May 23rd is a holiday and hence the university will be closed that day. STUDY: Chapter 1, Chapter 2 (excluding pgs 25-26). HOMEWORK PROBLEMS: 1. Let x , y R . Prove the following: If x < y then - y < - x . Use P1-P12. 2. Prove: | | x | - | y | | ≤ | x - y | for all x R . 3. Find all numbers x R such that | 3 x - 2 | = | 5 x + 4 | . 4. Let n N . Prove that n 2 < 2 n , for all n > 7. 5. Deﬁne the sequence of integers a 1 ,a 2

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Unformatted text preview: ,a 3 ,... as follows: a 1 = 3 a 2 = 6 a n = 5 a n-1-6 a n-2 + 2 , for n ≥ 3 Prove that a n = 1 + 2 n-1 + 3 n-1 for all n ∈ N . EXERCISES: You do not need to submit these questions but you should make sure that you are able to answer them; 1. Spivak # 1.3(i)-(iv); use P1-P9 for the proofs. 1.4(xiii)-(xiv), 1.5(i)-(iv); use P1-P12. 1.11(iii)-(v), 1.12(i)(v)(vi)(vii), 1.20, 2.1(i)(ii), 2.5(a). 2. Prove that (-1) x =-x for all x ∈ R . Use P1-P12. 3. Factor x 2-3 x + 2. Use P1-P12. 4. Show each of the following: (a) If | x | ≤ 2 then | x 3-x 2-cos ( x ) | ≤ 13. (b) If | x | ≥ 1 then | x | 3 ≥ | x | . (c) If | x | ≤ 1 2 then | x 1-x 2 | ≤ 2 3 . 5. Suppose that x 1 ,x 2 ,...,x n are real numbers. Prove that | x 1 + x 2 + ··· + x n | ≤ | x 1 | + | x 2 | + ··· + | x n | . 2...
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MATA37 2011summer - ,a 3 as follows a 1 = 3 a 2 = 6 a n = 5...

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