summation - CSCE 222 Discrete Structures for Computing...

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CSCE 222 Discrete Structures for Computing Sequences and Summations Dr. Hyunyoung Lee Based on slides by Andreas Klappenecker 1
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Sequences 2
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Sequences A sequence is a function from a subset of the set of integers (such as {0,1,2,. ..} or {1,2,3,. ..}) to some set S. We use the notation a n to denote the image of the integer n. We call a n a term of the sequence. We use the notations {a n } or (a n ) to denote sequences. 3
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Example Let us consider the sequence {a n }, where a n = 1/n. Thus, the sequence starts with {a 1 , a 2 , a 3 , a 4 , . .. } = {1, 1/2, 1/3, 1/4, . .. } Sequences find ubiquitous use in computer science. 4
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Geometric Progression A geometric progression is a sequence of the form: a, ar, ar 2 , ar 3 ,... where the initial term a and the common ratio r are real numbers. The geometric progression is a discrete analogue of an exponential function. 5
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Arithmetic Progression An arithmetic progression is a sequence of the form a, a+d, a+2d, a+3d, . .. where the initial term a and the common difference d are real numbers. An arithmetic function is a discrete analogue of a linear function dx+a. 6
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Strings The data type of a string is nothing but a sequence of finite length. 7
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Recurrence Relations 8
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Recurrence Relation A recurrence relation for a sequence {a n } expresses the term a n
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summation - CSCE 222 Discrete Structures for Computing...

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