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Learn to analyze and evaluate sequences in this lecture, including arithmetic and geometric progressions. Discover how to find formulas for different sequences and compute their sums. Understand countable sets and cardinality concepts. Homework assignments on pages 236-237.
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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