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Lecture 09: SEQUENCES Section 3.2

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## Lecture 09: SEQUENCES Section 3.2

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**Lecture 09: SEQUENCESSection 3.2**CS1050: Understanding and Constructing Proofs Spring 2006 Jarek Rossignac**Lecture Objectives**• Analyze/evaluate sequences**What is a sequence?**A function that maps an element n in the set {0,1,2…} or {1,2,3…} into the term an in an ordered set S. Notation {an} describes the sequence. For example {1/n} is {1, 1/2, 1/3,…}**What is an arithmetic progression?**{a+nd} a = initial term d = common difference Examples: {1, 3, 5, 7…} {(–1)n} = ?**What is a geometric progression?**{arn} a = initial term d = common ratio Examples: {1, 2, 4, 8…} {(–1)n} = ?**How to find the formula for a sequence**• 1, –1/2, 1/4, –1/8… an=? • 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, …? • 1, 8, 27… an=? • 3, 9, 27, 81… an=? • 1, 7, 25, 79, 241… an=?**How to compute sums of sequences?**∑k=0n(k) = (n+1)n/2 ∑k=0n(rk) = (rn+1–1)/(r–1) for r≠1 ∑k=0(xk) = 1/(1–x) for |x|<1**What is a countable set?**• A and B have the same cardinality if there is a bijection between them. • A set is countable is it has the same cardinality as the set of positive integers. • Positive rational numbers are countable • Real numbers are not**Assigned Homework**• Page 236-237: 9g and 28