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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 1 Homework, Page 739 Find the first six terms and the 100th term of the explicitly defined sequence. 1.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 2 Homework, Page 739 Find the first four terms and the eighth term of the recursively defined sequence. 5.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 3 Homework, Page 739 Find the first four terms and the eighth term of the recursively defined sequence. 9.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 4 Homework, Page 739 Determine whether the sequence converges or diverges. If it converges, give the limit.. 13.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 5 Homework, Page 739 Determine whether the sequence converges or diverges. If it converges, give the limit.. 17.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 6 Homework, Page 739 The sequences are arithmetic. Find: (a) the common difference, (b) the tenth term, (c) a recursive rule for the n th term, and (d) an explicit rule for the n th term. 21.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 7 Homework, Page 739 The sequences are geometric. Find: (a) the common ratio, (b) the eighth term, (c) a recursive rule for the n th term, and (d) an explicit rule for the n th term. 25.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 8 Homework, Page 739 29.The fourth and seventh terms of an arithmetic sequence are – 8 and 4, respectively. Find the first term and a recursive rule for the nth term.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 9 Homework, Page 739 Graph the sequence. 33.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 10 Homework, Page 739 37.The bungy-bungy tree grows an average of 2.3 cm per week. Write a sequence that represents the height of a bungy-bungy over the course of one-year if it is 7 meters tall today. Display the first four terms and the last two terms.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 11 Homework, Page 739 43.If the first two terms of a geometric sequence are negative, then so is the third. Justify your answer. True, if the first two numbers are negative, then the common ratio must be positive and the multiplication of a negative number by a positive number yields another negative number.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 12 Homework, Page 739 45.The first two terms of an arithmetic sequence are 2 and 8. The fourth term is A. 20 B. 26 C. 64 D. 128 E. 256

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 13 Homework, Page 739 47. A. 1 B. 4 C. 9 D. 12 E. 81

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 9.5 Series

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 15 Quick Review

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 16 Quick Review Solutions

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 17 What you’ll learn about Summation Notation Sums of Arithmetic and Geometric Sequences Infinite Series Convergences of Geometric Series … and why Infinite series are at the heart of integral calculus.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 18 Summation Notation

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 19 Sum of a Finite Arithmetic Sequence

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 20 Example Summing the Terms of an Arithmetic Sequence

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 21 Sum of a Finite Geometric Sequence

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 22 Example Summing the Terms of a Finite Geometric Sequence

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 23 Partial Sums Partial sums are the sums of a finite number of terms in an infinite sequence. In some instances, the partial sums approach a finite limit and the series is said to converge.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 24 Example Examining Partial Sums

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 25 Infinite Series

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 26 Sum of an Infinite Geometric Series

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 27 Example Summing Infinite Geometric Series

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 28 Homework Homework Assignment #34 Read Section 9.6 Page 749, Exercises: 1 – 45 (EOO)

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 9.6 Mathematical Induction

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 30 Quick Review

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 31 Quick Review Solutions

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 32 What you’ll learn about The Tower of Hanoi Problem Principle of Mathematical Induction Induction and Deduction … and why The principle of mathematical induction is a valuable technique for proving combinatorial formulas.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 33 The Tower of Hanoi Solution The minimum number of moves required to move a stack of n washers in a Tower of Hanoi game is 2 n – 1.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 34 Principle of Mathematical Induction Let P n be a statement about the integer n. Then P n is true for all positive integers n provided the following conditions are satisfied: 1. (the anchor) P 1 is true; 2. (inductive step) if P k is true, then P k+1 is true.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 35 Example Proving a Statement Using Mathematical Induction Use mathematical induction to prove the statement holds for all positive integers.

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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 9- 36 Induction and Deduction Induction - the process of using evidence from a particular example to draw conclusions about general principles Deduction - the process of using general principles to draw conclusions about specific examples

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One important application of infinite sequences is in representing “infinite summations.” Informally, if {a n } is an infinite sequence, then is an infinite.

One important application of infinite sequences is in representing “infinite summations.” Informally, if {a n } is an infinite sequence, then is an infinite.

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