SEQUENCES AND SERIES - Brooklyn Technical High School

1 CHAPTER6247 CHAPTERTABLE OFCONTENTS6-1 Sequences6-2 Arithmetic Sequences6-3 Sigma Notation6-4 Arithmetic Series6-5 geometric Sequences6-6 geometric Series6-7 Infinite SeriesChapter SummaryVocabularyReview ExercisesCumulative ReviewSEQUENCESAND SERIESWhen the Grant family purchased a computer for$1,200 on an installment plan, they agreed to pay $100 eachmonth until the cost of the computer plus interest had beenpaid. The interest each month was of the unpaid bal-ance. The amount that the Gant family still owed after eachpayment is a function of the number of months that havepassed since they purchased the computer. At the end of month 1:they owed (1,200 11,200 2100) dollars or(1,200 2100) 5$1,118.

2 At the end of month 2:they owed (1,118 11,118 2100) dollars or(1,118 2100) 5$1, month, interest is added to the balance from theprevious month and a payment of $100 is can express the monthly payments with the functiondefined as {(n,f(n))}. Let the domain be the set of positiveintegers that represent the number of months since the ini-tial :f(1) 51118f(2) general, for positive integers n:f(n) 5f(n 21) 2100 This pattern continues until (f(n21) ) isbetween 0 and 100, since the final payment would be (f(n21) ) this chapter we will study sequential functions, such asthe one described here, whose domain is the set of 8/12/08 1:50 PM Page 247A ball is dropped from height of 16 feet.

3 Each time that it bounces, it reaches aheight that is half of its previous height. We can list the height to which the ballbounces in order until it finally comes to numbers 8, 4, 2, 1, , ,form a sequenceis a set of num-bers written in a given order. We can list these heights as ordered pairs of num-bers in which each height is paired with the number that indicates its position inthe list. The set of ordered pairs would be:{(1, 8), (2, 4), (3, 2), (4, 1), (5, ), (6, ), (7, )}We associate each term of the sequence with the positive integer that specifiesits position in the ordered set. Therefore, a sequence is a special type of function that lists the height of the ballafter 7 bounces is shown on the graph at the the sequence can continue withoutend.

4 In this case, the domain is the set of terms of a sequence are often designated as a1,a2,a3,a4,a5, .. If thesequence is designated as the function f, then f(1) 5a1, f(2) 5a2, or in general:f(n) 5anMost SEQUENCES are sets of numbers that are related by some pattern thatcan be expressed as a formula. The formula that allows any term of a sequence ,except the first, to be computed from the previous term is called a SEQUENCES248 SEQUENCES and SeriesDEFINITIONA finite sequenceis a function whose domain is the set of integers {1, 2, 3, .. ,n}.DEFINITIONAn infinite sequenceis a function whose domain is the set of positive Bounce1 23456 7 Height (ft) 8/12/08 1:50 PM Page 248 For example, the sequence that lists the heights to which a ball bounceswhen dropped from a height of 16 feet is 8, 4, 2, 1.

5 In thissequence, each term after the first is the previous term. Therefore, for eachterm after the first,4 5,2 5,1 5, 5, 5, 5,..For , we can write the recursive definition:Alternatively, we can write the recursive definition as:for n$1A rule that designates any term of a sequence can often be determined fromthe first few terms of the the next three terms of the sequence 2, 4, 8, 16, .. a general expression for a recursive definition for the appears that each term of the sequence is a power of 2: 21,22,23,24, ..Therefore, the next three terms should be 25,26, and 27or 32, 64, and term is a power of 2 with the exponent equal to the number of theterm. Therefore, term is twice the previous term.

6 Therefore, for n .1, , for n$1, , 64, n .1,an52an21or for n$1, 2 Write the first three terms of the sequence (1)21 52a253(2)21 55a353(3)21 58 AnswerThe first three terms of the sequence are 2, 5, and ( )12( )12(1)12(2)12(4)12(8) 8/12/08 1:50 PM Page 249 SEQUENCES and the Graphing CalculatorWe can use the sequence function on the graphing calculator to view a sequencefor a specific range of terms. Evaluate a sequencein terms of variablefrom abeginning term to an ending term. That is:seq( sequence ,variable,beginning term,ending term)For example, we can examine the sequence on the view the first 20 terms of the sequence :ENTER:21 20 DISPLAY:Note:We use the variable Xinstead of nto enter the sequence .

7 We can also usethe left and right arrow keys to examine the terms of the About said that sequence of numbers in which each term equals half of the previous termis a finite sequence . Randi said that it is an infinite sequence . Who is correct? Justify said that if an53n21, then an115an13. Do you agree with Jacob? Explain whyor why said that if an52n, then an1152n11. Do you agree with Carlos? Explain why orwhy SkillsIn 3 18, write the first five terms of each {3 6 11 18 27 (X2+2,X,1,20)ENTER),,X,T, ,n, x2X,T, ,n5 LIST2ndan5n212250 SEQUENCES and 8/12/08 1:50 PM Page 250In 19 30 an algebraic expression that represents anfor each the ninthterm of each , 4, 6, 8, .. , 6, 9, 12, .. , 4, 7, 10.}

8 , 9, 27, 81, .. , 6, 3, , .. , 9, 11, 13, .. ,8i,6i,4i,..26.,,,,..27.,,,,.. , 5, 10, 17, .. ,22, 3,24,.. ,,,2,..In 31 39, write the first five terms of each ,an= , , , , , , , ,an5 Applying has started an exercise program. The first day he worked out for 30 minutes. Each dayfor the next six days, he increased his time by 5 the sequence for the number of minutes that Sean worked out for each of theseven a recursive definition for this wants to increase her vocabulary. On Monday she learned the meanings of four newwords. Each other day that week, she increased the number of new words that she learnedby the sequence for the number of new words that Sherri learned each day for a recursive definition for this is trying to lose weight.

9 She now weighs 180 pounds. Every week for eight weeks, shewas able to lose 2 Julie s weight for each a recursive definition for this 1, 2008, was a the dates for each Tuesday in January of that a recursive definition for this started a new job with a weekly salary of $400. After one year, and for each year thatfollowed, his salary was increased by 10%. Hui left this job after six the weekly salary that Hui earned each a recursive definition for this of the most famous SEQUENCES is the Fibonacci sequence . In this sequence ,a151,a251, and for ,an=an221an21. Write the first ten terms of this !3! 8/12/08 1:50 PM Page 251 Hands-On ActivityThe Tower of Hanoi is a famous problem that has challenged problem solvers throughout the tower consists of three pegs.

10 On one peg there are a number of disks of different sizes, stackedaccording to size with the largest at the bottom. The task is to move the entire stack from one pegto the other side using the following rules: Only one disk may be moved at a time. No disk may be placed on top of a smaller that a move consists of taking the top disk from one peg and placing it on another a stack of different-sized coins to model the Tower of Hanoi. What is the smallest num-ber of moves needed if there disks? disks? disks? a recursive definition for the sequence described in Exercise 1,a set of positive odd numbers, 1, 3, 5, 7, .. , is a sequence . The first term,a1,is 1 and each term is 2 greater than the preceding term.