Copyright © 2011 Pearson Education, Inc. Slide

Copyright © 2011 Pearson Education, Inc. Slide Chapter 11: Further Topics in Algebra 11.1Sequences and Series 11.2Arithmetic Sequences and Series 11.3Geometric Sequences and Series 11.4 Counting Theory 11.5The Binomial Theorem 11.6 Mathematical Induction 11.7Probability

Copyright © 2011 Pearson Education, Inc. Slide Sequences Sequences are ordered lists generated by a function, for example f(n) = 100n

Copyright © 2011 Pearson Education, Inc. Slide f (x) notation is not used for sequences. Write Sequences are written as ordered lists a 1 is the first element, a 2 the second element, and so on 11.1 Sequences A sequence is a function that has a set of natural numbers (positive integers) as its domain.

Copyright © 2011 Pearson Education, Inc. Slide Graphing Sequences The graph of a sequence, a n, is the graph of the discrete points (n, a n ) for n = 1, 2, 3, … Example Graph the sequence a n = 2n. Solution

Copyright © 2011 Pearson Education, Inc. Slide Sequences A sequence is often specified by giving a formula for the general term or nth term, a n. Example Find the first four terms for the sequence Solution

Copyright © 2011 Pearson Education, Inc. Slide Sequences A finite sequence has domain the finite set {1, 2, 3, …, n} for some natural number n. Example 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 An infinite sequence has domain {1, 2, 3, …}, the set of all natural numbers. Example 1, 2, 4, 8, 16, 32, …

Copyright © 2011 Pearson Education, Inc. Slide Convergent and Divergent Sequences A convergent sequence is one whose terms get closer and closer to a some real number. The sequence is said to converge to that number. A sequence that is not convergent is said to be divergent.

Copyright © 2011 Pearson Education, Inc. Slide Convergent and Divergent Sequences Example The sequence converges to 0. The terms of the sequence 1, 0.5, , 0.25, … grow smaller and smaller approaching 0. This can be seen graphically.

Copyright © 2011 Pearson Education, Inc. Slide Convergent and Divergent Sequences Example The sequence is divergent. The terms grow large without bound 1, 4, 9, 16, 25, 36, 49, 64, … and do not approach any one number.

Copyright © 2011 Pearson Education, Inc. Slide Sequences and Recursion Formulas A recursion formula or recursive definition defines a sequence by –Specifying the first few terms of the sequence –Using a formula to specify subsequent terms in terms of preceding terms.

Copyright © 2011 Pearson Education, Inc. Slide Using a Recursion Formula Example Find the first four terms of the sequence a 1 = 4; for n >1, a n = 2a n Solution We know a 1 = 4. Since a n = 2a n-1 + 1

Copyright © 2011 Pearson Education, Inc. Slide Applications of Sequences Example The winter moth population in thousands per acre in year n, is modeled by for n > 2 (a)Give a table of values for n = 1, 2, 3, …, 10 (b)Graph the sequence.

Copyright © 2011 Pearson Education, Inc. Slide Applications of Sequences Solution (a) (b) Note the population stabilizes near a value of 9.7 thousand insects per acre. n anan n78910 anan

Copyright © 2011 Pearson Education, Inc. Slide Series and Summation Notation S n is the sum a 1 + a 2 + …+ a n of the first n terms of the sequence a 1, a 2, a 3, ….  is the Greek letter sigma and indicates a sum. The sigma notation means add the terms a i beginning with the 1 st term and ending with the nth term. i is called the index of summation.

Copyright © 2011 Pearson Education, Inc. Slide Series and Summation Notation A finite series is an expression of the form and an infinite series is an expression of the form.

Copyright © 2011 Pearson Education, Inc. Slide Series and Summation Notation Example Evaluate (a)(b) Solution (a) (b)

Copyright © 2011 Pearson Education, Inc. Slide Series and Summation Notation Summation Properties If a 1, a 2, a 3, …, a n and b 1, b 2, b 3, …, b n are two sequences, and c is a constant, then, for every positive integer n, (a)(b) (c)

Copyright © 2011 Pearson Education, Inc. Slide Series and Summation Notation Summation Rules

Copyright © 2011 Pearson Education, Inc. Slide Series and Summation Notation Example Use the summation properties to evaluate (a) (b) (c) Solution (a)

Copyright © 2011 Pearson Education, Inc. Slide Series and Summation Notation Solution (b) (c)