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8.1 Sequences Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Quick Review Let f ( x) x . Find the values of f . x4 1. f (5) 2. f (-1) Evaluate the expression a n 1 d for the given values of a, n, and d . 3. a -2, n 2, d 3 4. a 1, n 2, d 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 2 Quick Review n -1 Evaluate the expression ar for the given values of a, r , and n. 1 5. a , r 2, n 3 2 6. a 2, r 1.5, n 4 Find the value of the limit. 2x 2 7. lim 4x x 1 sin 4 x 8. lim x 2 x 2 x 0 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 3 Quick Review Solutions x . Find the values of f . x4 5 1. f (5) 9 1 2. f (-1) 3 Let f ( x) Evaluate the expression a n 1 d for the given values of a, n, and d . 3. a -2, n 2, d 3 1 4. a 1, n 2, d 2 -5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 4 Quick Review Solutions n -1 Evaluate the expression ar for the given values of a, r , and n. 1 5. a , r 2, n 3 2 2 6. a 2, r 1.5, n 4 -6.75 Find the value of the limit. 2x 2 1 7. lim 4x x 1 2 sin 4 x 8. lim 4 x 2 x 2 x 0 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 5 What you’ll learn about Defining a Sequence Arithmetic and Geometric Sequences Graphing a Sequence Limit of a Sequence …and why Sequences arise frequently in mathematics and applied fields. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 6 Defining a Sequence A sequence a is a list of numbers written in an explicit order. n For example: a a , a , a ,..., a ,... , where a is the first term n 1 2 3 n 1 and a is the nth term of the sequence. n Let a , a , a ,..., a ,... be a function with domain the set of positive 1 2 3 n integers and range a , a , a ,..., a ,.... If the domain is finite, then 1 2 3 n the sequence is a finite sequence. If the domain is infinite, then the sequence is an infinite sequence. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 7 Example Defining a Sequence Explicitly Find the first four terms and the 100th term of the sequence a n 1 where a . n 2 Set n equal to 1, 2, 3, 4, and 100. n n 2 1 1 1 2 3 1 1 2 2 6 1 11 1 18 1 10, 002 1 a 1 2 2 a 2 a 3 a a 4 100 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 8 Example Defining a Sequence Recursively Find the first three terms and the seventh term for the sequence defined recursively by the conditions: b 4 and b b 2 for all n 2. 1 n 1 n Since b 4 and b b 2, you can find b 2, b 0, and b 8. 1 n n 1 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall 3 7 Slide 8- 9 Arithmetic Sequence A sequence a is an arithmetic sequence if it can be written in the n form a, a d , a 2d ,..., a n -1 d ,... for some constant d . The number d is the common difference. Each term in an arithmetic sequence can be obtained recursively from its preceeding term by adding d : a a d for all n 2. n n 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 10 Example Defining Arithmetic Sequences Given the arithmetic sequence: 3, 1, 5, 9, ... find (a) the common difference, (b) the ninth term, (c) a recursive rule for the nth term, (d) an explicit rule for the nth term. (a) The common difference between terms is 4. (b) a 3 9 1 (4) 29. 9 (c) a 3 and a a 4 for all n 2. 1 n n 1 (d) a 3 (n 1)(4) n 4n 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 11 Geometric Sequence A sequence a is a geometric sequence if it can be written in the n form a, a r , a r ,..., a r ,... for some nonzero constant r. 2 n 1 The number r is the common ratio. Each term in a geometric sequence can be obtained recursively from its preceeding term by multiplying by r: a a r for all n 2. n n 1 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 12 Example Defining Geometric Sequences For the geometric sequence 1, 3, 9, 27, ..., find (a) the common ratio, (b) the tenth term, (c) a recursive rule for the nth term, (d) an explicit rule for the nth term. (a) The common ratio is 3. (b) a (1) (3) 19683 9 10 (c) The sequence is defined recursively by a 1 and a 3 a 1 n n 1 for n 2. (d) The sequence is defined explicitly by a 1 3 3 . n1 n1 n Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 13 Example Constructing a Sequence The second and fifth term of a geometric sequence are -6 and 48, respectively. Find the first term, common ratio and an explicit rule for the nth term. 4 ar 48 ar 6 1 1 r 8 r 2 Then a r 6 means that a 3. 3 1 1 The sequence is defined explicitly: a (3) 2 (1) n 1 n Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall n 1 3 2 n 1 Slide 8- 14 Example Graphing a Sequence Using Parametric Mode Draw a graph of the sequence a with a 1 n n n n 1 , n 1, 2,... . n T 1 Let X =T,Y =( 1) , and graph in dot mode. Set T 1, T T 20, and T 1. Choose X 0, X 20, X 2, Y = 2, T 1T 1T max min step min max scl min Y 2, Y 1, and draw the graph. max scl Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 15 Example Graphing a Sequence Using Sequence Graphing Mode Graph the sequence defined recursively by b 4 and 1 b b 2 for all n 2. n n -1 Set the graph in sequence graphing mode and dot mode. Replace b by u (n). Select nMin 1, u (n) u (n 1) 2, n and u (nMin) 4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 16 Example Graphing a Sequence Using Sequence Graphing Mode Set nMin 1, nMax 10, PlotStart 1, PlotStep 1, and graph in the 0,10 by -5,25 viewing window Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 17 Limit Let L be a real number. The sequence a has limit L as n approaches n if, given any positive number , there is a positive number M such that for all n M we have a - L . n We write lim a L and say that the sequence converges to L. n n Sequences that do not have limits diverge. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 18 Properties of Limits If L and M are real numbers and lim a L and lim b M , then n n n n 1. Sum Rule: lim a b L M n n n 2. Difference Rule: lim a b L M n n n 3. Product Rule: lim a b L M n n n 4. Constant Multiple Rule: lim ca c L n n a L 5. Quotient Rule: lim , M 0 b M n n n Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 19 Example Finding the Limit of a Sequence Determine whether the sequence converges or diverges. If it converges, find its limit. a n 2n 1 n Analytically, using the Properties of Limits: lim n 2n 1 1 lim 2 n n n 1 lim(2) lim n 20 2 n n Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 20 The Sandwich Theorem for Sequences If lim a lim c L and if there is an integer N for which n n n n a b c for all n N , then lim b L. n n n n n Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall Slide 8- 21 Absolute Value Theorem Consider the sequence a . If lim a 0, then lim a 0. n n n Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall n n Slide 8- 22