ICS 253: Discrete Structures I King Fahd University of Petroleum & Minerals Information & Computer Science Department ICS 253: Discrete Structures I Basic Structures: Sets, Functions, Sequences and Sums

Section 2.4: Sequences and Summations A sequence is a function from a subset of the set of integers (usually either the set {0, 1, 2, . . .} or the set {1, 2, 3, . . .}) to a set S. We use the notation an to denote the image of the integer n. We call an a term of the sequence. The notation {an} is used to describe the sequence

Notation A geometric progression is a sequence of the form a, ar, ar 2 ,..., ar n ,... where the initial term a and the common ratio r are real numbers. A geometric progression is a discrete analogue of the exponential function f (x) = ar x .

Notation An arithmetic progression is a sequence of the form a, a + d, a + 2d, . . . , a + n d, . . . where the initial term a and the common difference d are real numbers. An arithmetic progression is a discrete analogue of the linear function ……

Examples Q2 pp 160: 2. What is the term a8 of the sequence {an} if an equals a) 2n – l? b) 7? c) 1 + (–1)n ? d) –(–2)n?

Sequence Generalization The problem is how to generalize a sequence from its first few terms. Examples 1, 1/2, 1/4, 1/8, 1/16, … 1, 3, 5, 7, 9, … 1, –1, 1, –1 , 1, … 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, … 5, 11, 17, 23, 29, 35, 41, 47, 53, 59, …

A Table to Memorize! nth term First 10 terms n2 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, ... n3 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, ... n4 1, 16, 81, 256, 625, 1296, 2401, 4096, 6561, 10000, ... 2n 2, 4, 8, 16, 32, 64, 128, 256 , 512, 1024, ... 3n 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, 59049, ... n! 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800,...

More Examples 1, 7, 25, 79, 241, 727, 2185, 6559, 19681, 59047, … Q10 (b,c) pp 161: For each of these lists of integers, provide a simple formula or rule that generates the terms of an integer sequence that begins with the given list. Assuming that your formula or rule is correct, determine the next three terms of the sequence. b) 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, ... c) 1, 10, 11, 100, 101, 110, 111, 1000, 1001 , 1010, 1011, . . .

b) 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, ... c) 1, 10, 11, 100, 101, 110, 111, 1000, 1001 , 1010, 1011, . . .

Summations Important rule: j : index of summation, can be replaced by any arbitrary variable m: lower limit n: upper limit Important rule:

Examples Express the sum of the first 100 terms of the sequence {an}, where an = 1/n for n = 1, 2, 3, …

Index Changes in the Summation Consider the summation and assume that we want the index to start from 0 to n – 1 rather than 1 to n. How do we change the index?

Theorem 1 If a and r are real numbers and r  0, then Proof

Examples

Some Useful Summations

More Examples Find Let x be a real number with |x|<1. Find

More Examples Q10 (a) pp 161: For each of these lists of integers, provide a simple formula or rule that generates the terms of an integer sequence that begins with the given list. Assuming that your formula or rule is correct, determine the next three terms of the sequence. 3, 6, 11, 18, 27, 38, 51, 66, 83, 102, ...

Cardinality Definition: The sets A and B have the same cardinality if and only if there is a one-to-one correspondence from A to B. Definition: A set that is either finite or has the same cardinality as the set of positive integers is called countable. A set that is not countable is called uncountable. When an infinite set S is countable, we denote the cardinality of S by  0 (where 0 is aleph, the first letter of the Hebrew alphabet). We write |S| = 0 and say that S has cardinality "aleph null."

Cardinality Question: What do we need to do to find whether a set is countable or not? Example 1: Show that the set of odd positive integers is a countable set.

Example 2: Show that the set of all integers is countable. Cardinality Example 2: Show that the set of all integers is countable. F(n) = 2n -1

Cardinality Example 3: Show that the set of positive rational numbers is countable.

Cardinality Example 4: Show that the set of real numbers is an uncountable set.