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Warm-Up • Find the first four terms of the sequence given by an 3n 2 Find the first four terms of the n sequence given by an 3 (1) a1 31 2 1 a1 3 (1)1 3 1 2 a2 3 2 2 4 a2 3 (1)2 3 1 4 a3 33 2 7 a3 3 (1)3 3 1 2 a4 3 4 2 10 a4 3 (1)4 3 1 4 Copyright © 2007 Pearson Education, Inc. Slide 8-2 Chapter 8: Sequences and Series 2015 Copyright © 2007 Pearson Education, Inc. Slide 8-3 Chapter 8: Sequences, Series, and Probability 8.1 Sequences and Series 8.2 Arithmetic Sequences and Series 8.3 Geometric Sequences and Series 8.4 Mathematical Induction 8.5 The Binomial Theorem Copyright © 2007 Pearson Education, Inc. Slide 8-4 8.1 Sequences A sequence is a function that has a set of natural numbers as its domain. • • • f (x) notation is not used for sequences. Write an f (n) Sequences are written as ordered lists a1 , a2 , a3 , ... • a1 is the first element, a2 the second element, and so on Copyright © 2007 Pearson Education, Inc. Slide 8-6 8.1 Sequences A sequence is often specified by giving a formula for the general term or nth term, an. Example Find the first four terms for the sequence n 1 an n2 Solution a1 2 / 3, Copyright © 2007 Pearson Education, Inc. a2 3 / 4, a3 4 / 5, a4 5 / 6 Slide 8-7 8.1 Graphing Sequences The graph of a sequence, an, is the graph of the discrete points (n, an) for n = 1, 2, 3, … Example Graph the sequence an = 2n. Solution Copyright © 2007 Pearson Education, Inc. Slide 8-8 8.1 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 © 2007 Pearson Education, Inc. Slide 8-9 8.1 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 © 2007 Pearson Education, Inc. Slide 8-10 8.1 Convergent and Divergent Sequences Example : 1 Find the first 5 terms of the sequence an . n Is the sequence convergent or divergent? Copyright © 2007 Pearson Education, Inc. Slide 8-11 8.1 Convergent and Divergent Sequences 1 Solution: The sequence an converges to 0. n The terms of the sequence 1, 0.5, 0.33.., 0.25, … grow smaller and smaller approaching 0. This can be seen graphically. Copyright © 2007 Pearson Education, Inc. Slide 8-12 8.1 Convergent and Divergent Sequences Example : Find the first 6 terms of the sequence an n . 2 Is the sequence convergent or divergent? Copyright © 2007 Pearson Education, Inc. Slide 8-13 8.1 Convergent and Divergent Sequences Solution: The sequence an n2 is divergent. The terms grow large without bound 1, 4, 9, 16, 25, 36, 49, 64, … and do not approach any one number. Copyright © 2007 Pearson Education, Inc. Slide 8-14 8.1 Convergent and Divergent Sequences 2n n 3 an 2 3n 2n 5 2 Example Is the sequence convergent or divergent? Solution: The sequence converges to 2/3 Copyright © 2007 Pearson Education, Inc. Slide 8-15 Finding Terms of a Sequence • Write out the first five terms of the sequence given by (1) n an 2n 1 Solution: (1)1 1 a1 1 2 1 1 2 1 (1)2 1 1 a2 2 2 1 4 1 3 Copyright © 2007 Pearson Education, Inc. (1)3 1 1 a3 2 3 1 6 1 5 (1)4 1 1 a4 2 4 1 8 1 7 (1)5 1 1 a5 2 5 1 10 1 9 Slide 8-17 Finding the nth term of a Sequence • Write an expression for the apparent nth term (an) of each sequence. • a. 1, 3, 5, 7, … b. 2, 5, 10, 17, … Solution: a. n: 1 2 3 4 . . . n terms: 1 3 5 7 . . . an Apparent pattern: Each term is 1 less than twice n, which implies that an 2n 1 Copyright © 2007 Pearson Education, Inc. b. n: 1 2 3 4 … n terms: 2 5 10 17 … an Apparent pattern: Each term is 1 more than the square of n, which implies that an n2 1 Slide 8-18 Additional Example • Write an expression for the apparent nth term of the sequence: 2 3 4 5 , , , ,... 1 2 3 4 Solution: n : 1 2 3 4 ... n 2 3 4 5 terms: ... an 1 2 3 4 Apparent pattern: Each term has a n 1 numerator that is 1 greater than its an n denominator, which implies that Copyright © 2007 Pearson Education, Inc. Slide 8-19 Factorial Notation • If n is a positive integer, n factorial is defined by n! 1 2 3 4... (n 1) n As a special case, zero factorial is defined as 0! = 1. Here are some values of n! for the first several nonnegative integers. Notice that 0! is 1 by definition. 0! 1 3! 1 2 3 6 1! 1 2! 1 2 2 4! 1 2 3 4 24 5! 1 2 3 4 5 120 The value of n does not have to be very large before the value of n! becomes huge. For instance, 10! = 3,628,800. Copyright © 2007 Pearson Education, Inc. Slide 8-20 Finding the Terms of a Sequence Involving Factorials • List the first five terms of the sequence given by 2n an n! Begin with n = 0. 22 4 a2 2 2! 2 20 1 a0 1 0! 1 23 8 4 a3 3! 6 3 21 2 a1 2 1! 1 24 16 2 a4 4! 24 3 Copyright © 2007 Pearson Education, Inc. Slide 8-21 Evaluating Factorial Expressions • Evaluate each factorial expression. Make sure you use parentheses when necessary. a. b. 2! 6! c. n! 8! 2! 6! 3! 5! (n 1)! a. 8! 1 2 3 4 5 6 7 8 7 8 28 2! 6! 1 2 1 2 3 4 5 6 2 b. 2! 6! 1 2 1 2 3 4 5 6 6 2 3! 5! 1 2 3 1 2 3 4 5 3 c. n! 1 2 3...(n 1) n n (n 1)! 1 2 3...(n 1) Copyright © 2007 Pearson Education, Inc. Slide 8-22 Additional Example • Write an expression for the apparent nth term of the sequence: 2 2 23 2 4 25 1, 2, , , , ,... 2 6 24 120 Solution: n: 1 2 3 4 5 6 ... n 2 2 23 2 4 25 terms: 1, 2, , , , ... an 2 6 24 120 Apparent pattern: Each term has a 2n 1 numerator that is 1 greater than its an n 1! denominator, which implies that Copyright © 2007 Pearson Education, Inc. Slide 8-23 Have you ever seen this sequence before? • 1, 1, 2, 3, 5, 8 … • Can you find the next three terms in the sequence? • Hint: 13, • 21, 34 • Can you explain this pattern? Copyright © 2007 Pearson Education, Inc. Slide 8-24 The Fibonacci Sequence • Some sequences are defined recursively. To define a sequence recursively, you need to be given one or more of the first few terms. A well-known example is the Fibonacci Sequence. • The Fibonacci Sequence is defined as follows: a0 1, a1 1, ak ak 2 ak 1, where k 2 Write the first six terms of the Fibonacci Sequence: a32 a31 a1 a2 1 2 3 a0 1 a42 a41 a2 a3 2 3 5 a 1 1 a22 a21 a0 a1 1 1 2 Copyright © 2007 Pearson Education, Inc. a52 a51 a3 a4 3 5 8 Slide 8-25 Example • Write the first five terms of the recursively defined sequence: a1 5, ak 1 ak 3 Solution: 5, 8, 11, 14, 17 Copyright © 2007 Pearson Education, Inc. Slide 8-26 Homework • Day 1: Pg. 563 1-9odd, 21-23odd, 35-69 odd • Day 2: 71-81 odd, 91-103 odd Copyright © 2007 Pearson Education, Inc. Slide 8-27 HWQ Write an expression for the apparent nth term of the sequence. 1 3 7 15 31 1 ,1 ,1 ,1 ,1 ,... 2 4 8 16 32 Copyright © 2007 Pearson Education, Inc. Slide 8-28 8.1 Day 2 Series 2015 Copyright © 2007 Pearson Education, Inc. Slide 8-29 Summation Notation • Definition of Summation Notation The sum of the first n terms of a sequence is represented by n a i 1 n a1 a2 a3 a4 ... an Where i is called the index of summation, n is the upper limit of summation and 1 is the lower limit of summation. Copyright © 2007 Pearson Education, Inc. Slide 8-30 8.1 Series and Summation Notation A finite series is an expression of the form n Sn a1 a2 a3 ... an ai i 1 and an infinite series is an expression of the form S a1 a2 a3 ... an ... ai . i 1 Copyright © 2007 Pearson Education, Inc. Slide 8-31 Summation Notation for Sums • Find each sum. a. 5 3i i 1 b. 6 (1 k k 3 2 ) c. 8 1 i 0 i ! Solution: 5 a. 3i 3(1) 3(2) 3(3) 3(4) 3(5) i 1 3(1 2 3 4 5) or 3 6 9 12 15 45 Copyright © 2007 Pearson Education, Inc. Slide 8-32 Solutions continued 6 b. (1 k 2 ) (1 32 ) (1 42 ) (1 52 ) (1 62 ) k 3 10 17 26 37 90 c. 8 1 1 1 1 1 1 1 1 1 1 0! 1! 2! 3! 4! 5! 6! 7! 8! i 0 i ! 1 1 1 1 1 1 1 11 2 6 24 120 720 5040 40320 2.71828 Notice that this summation is very close to the irrational number e 2.718281828 . It can be shown that as more terms of the sequence whose nth term is 1/n! are added, the sum becomes closer and closer to e. Copyright © 2007 Pearson Education, Inc. Slide 8-33 8.1 Series and Summation Notation Summation Properties If a1, a2, a3, …, an and b1, b2, b3, …, bn are two sequences, and c is a constant, then for every positive integer n, n (a) n c nc (b) i 1 i 1 n (c) n i c ai i 1 n (a b ) a b i 1 Copyright © 2007 Pearson Education, Inc. ca n i i i 1 i i 1 i Slide 8-34 8.1 Series and Summation Notation Summation Rules n(n 1) i 1 2 ... n 2 i 1 n n(n 1)(2n 1) 2 2 2 2 i 1 2 ... n 6 i 1 n 2 2 n ( n 1) 3 3 3 3 i 1 2 ... n 4 i 1 n These summation rules can be proven by mathematical induction. Copyright © 2007 Pearson Education, Inc. Slide 8-35 8.1 Series and Summation Notation Example Use the summation properties to 22 40 14 evaluate (a) 5 (b) 2i (c) (2i 2 3) i1 i 1 i 1 Solution 40 (a) 5 40(5) 200 i1 Copyright © 2007 Pearson Education, Inc. Slide 8-36 8.1 Series and Summation Notation 22 (b) 2i 14 (c) i 1 (b) (c) 2 (2 i 3) i 1 22(22 1) 2i 2 i 2 506 2 i 1 i 1 22 22 14 14 14 14 14 i 1 i 1 i 1 i 1 i 1 2 2 2 (2 i 3) 2 i 3 2 i 3 14(14 1)(2 14 1) 2 14(3) 1988 6 Copyright © 2007 Pearson Education, Inc. Slide 8-37 Homework • Day 1: Pg. 563 1-9odd, 21-23odd, 35-69 odd • Day 2: 71-81 odd, 91-103 odd Copyright © 2007 Pearson Education, Inc. Slide 8-38