i J i s k

O In Chapters 1 -4 ,you modeled data using various types of functions.

O In Chapter 10, you will:

Relate sequences and functions.

Represent and calculate sums of series with sigma notation.

Use arithmetic and geometric sequences and series.

Prove statements by using mathematical induction.

Expand powers by using the Binomial Theorem.

O MARCHING BAND Sequences and series can be used to predict patterns. For example, arithmetic sequences can be used to determine the number of band members in a specified row of a pyramid formation.

PREREAD Use the text on this page to predict the organization of Chapter 10.

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Practice Worksheets

Get Ready for the ChapterDiagnose Readiness You have two options for checking Prerequisite Skills.

1 Textbook Option Take the Quick Check below.QuickCheck

Expand each binomial. (Prerequisite Skill)

1. (x + 3)3 2. ( 2 x - 1)4

Use the graph of each function to describe its end behavior. Support the conjecture numerically. (Lesson 1-3)

Sketch and analyze the graph of each function. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing. (Lesson 3-1)

5. f(x) = 3~x 6. r {>0 = 5 - *

7. h(x) = 0.1x + 2 8. k(x) = - 2*

Evaluate each expression. (Lesson 3-2)

9. log2 16 10. Iog1010 11. log61216

12. MUSIC The table shows the type and number of CDs that Adam and Lindsay bought. Write and solve a system of equations to determine the price of each type of CD. (Lesson 6 -1 )

Adam 2 5 49

Lindsay 3 4 56

2 Online Option Take an online self-check Chapter Readiness Quiz at connectED.mcaraw-hill.com.

NewVocabularyEnglish Espanol

sequence p. 590 sucesionterm p. 590 termino

finite sequence p. 590 sucesion finitainfinite sequence p. 590 sucesion infinita

recursive sequence p. 591 sucesion recursivaexplicit sequence p. 591 sucesion explicita

Fibonacci sequence p. 591 sucesion de Fibonacciconverge p. 592 converge

diverge p. 592 divergeseries p. 593 serie

finite series p. 593 serie finitanth partial sum p. 593 suma parcial enesima

infinite series p. 593 serie infinitasigma notation p. 594 notacion de suma

arithmetic sequence p. 599 sucesion aritmeticacommon difference p. 599 diferencia comun

arithmetic series p. 602 serie aritmeticacommon ratio p. 608 razon comun

geometric means p. 611 medios geometricosgeometric series p. 611 serie geometrica

binomial coefficients p. 628 coeficientes binomialespower series p. 636 serie de potencias

ReviewVocabularyexponential function p. 158 funciones exponenciales a function in which the base is a constant and the exponent is a variable

Exponential Growth Function

589

You used functions to generate ordered pairs and used graphs to analyze end behavior.(Lesson 1-1 and 1-3)

4 Investigate several different types of sequences.

2 Use sigma notation to represent and calculate sums of series.

Khari developed a Web site where students at her high school can post their own social networking Web pages. A student at the high school is given a free page if he or she refers the Web site to five friends. The site starts with one page created by Khari, who in turn, refers five friends that each create a page. Those five friends refer five more people each, all of whom develop pages, and so on.

j s j NewVocabulary"rt: sequence

termfinite sequence infinite sequence recursive sequence explicit sequence Fibonacci sequence converge diverge series finite series nth partial sum infinite series sigma notation

590 Lesson 10-1

1 Sequences In mathematics, a sequence is an ordered list of numbers. Each number in the

sequence is known as a term. A finite sequence, such as 1, 3, 5, 7, 9 ,11 , contains a finite number of terms. An infinite sequence, such as 1, 3, 5, 7, ..., contains an infinite number of terms.

Each term of a sequence is a function of its position. Therefore, an infinite sequence is a function whose domain is the set of natural numbers and can be written as/ (l) = fl1,/(2) = a2, f ( 3) = a3, ..., f ( n ) = a n, ..., where an denotes the nth term. If the domain of the function is only the first n natural numbers, the sequence is finite.

Infinitely many sequences exist with the same first few terms. To sufficiently define a unique sequence, a formula for the ?zth term or other information must be given. When defined explicitly, an explicit formula gives the nth term a n as a function of n.

msm Find Terms of Sequencesa. Find the next four terms of the sequence 2, 7 ,1 2 ,1 7 , . . . .

The nth term of this sequence is not given. One possible pattern is that each term is 5 greater than the previous term. Therefore, a sample answer for the next four terms is 22 ,2 7 ,3 2 , and 37.

b. Find the next four terms of the sequence 2, 5 ,1 0 ,1 7 , . . . .

The nth term of this sequence is not given. If we subtract each term from the term that follows, we start to see a possible pattern.

a 2 i = 5 2 or 3 a 3 a2 = 10 5 or 5 17 - 10 or 7

It appears that each term is generated by adding the next successive odd number. However, looking at the pattern, it may also be determined that each term is 1 more than each perfect square, or a n = n2 + 1. Using either pattern, a sample answer for the next four terms is 26, 37, 50, and 65.

C. Find the first four terms of the sequence given by an = 2n(1)".Use the explicit formula given to find a n for n = 1, 2, 3, and 4.

fll = 2 1 (l ) 1 or - 2

a , = 2 - 3 - ( - l ) 3 or - 6

a2 = 2 2 (l ) 2 or 4 n = 2

n = A3 z. j - l) ui u ii o 2 4 (1) or J

The first four terms in the sequence are 2, 4, 6, and 8.

GuidedPracticeFind the next four terms of each sequence.

1A. 3 2 ,1 6 ,8 ,4 , . . . 1B. 1 ,2 ,4 ,7 ,1 1 ,1 6 ,2 2 , . . .

1C. Find the first four terms of the sequenc e given by an 10.

Sequences can also be defined recursively. Recursively defined sequences give one or more of the first few terms and then define the terms that follow using those previous terms. The formula defining the nth term of the sequence is called a recursive formula or a recurrence relation.

StudyTipNotation The term denoted a represents the nth term of a sequence. The term denoted a _ i represents the term immediately before a. The term a _ 2 represents the term two terms before a.

>Recursively Defined Sequences

Find the fifth term of the recursively defined sequence a 1 = 1, a n = a n _ 1 + 2n 1, where n > 2.

Since the sequence is defined recursively all the terms before the fifth term must be found first. Use the given first term, a 1 = 2, and the recursive formula for a n.

a 2 = a 2 _ i + 2(2) - 1 n= 2

= a + 3 Simplify,

= 2 + 3 or 5 a , = 2

^3 = #3 _ i + 2(3) 1 n = 3

= a 2 + 5 or 10 a2 = 5

a i = a i _ 1 + 2(4) - 1 n = 4

= a3 + 7 or 17 a 3 = 10

5 = f l 5 - 1 + 2(5) - 1 /? = 5

= a4 + 9 or 26 a4 = 17

p GuidedPracticeFind the sixth term of each sequence.

2A. f l j = 3, a n = (2 )an _ v n > 2 2B. a 1 = 8, an = 2 a n _ 1 7 ,n > 2

The Fibonacci sequence describes many patterns found in nature. This sequence is often defined recursively.

Fibonacci Sequence

Real-WorldLinkAlong with being found in flower petals, sea shells, and the bones in a human hand, Fibonacci sequences can also be found in pieces of art, music, poetry, and architecture.

Source: Universal Principles of Design

NATURE Suppose that when a plant first starts to grow, the stem has to grow for two months before it is strong enough to support branches. At the end of the second month, it sprouts a new branch and will continue to sprout one new branch each month. The new branches also each grow for two months and then start to sprout one new branch each month. If this pattern continues, how many branches will the plant have after 10 months?

During the first two months, there will only be one branch, the stem. At the end of the second month, the stem will produce a new branch, making the total for the third month two branches. The new branch will grow and develop two months before producing a new branch of its own, but the original branch will now produce a new branch each month.

Branches Months

591

WatchOut!Notation The first term of a sequence is occasionally denoted as a0 When this occurs, the domain of the function describing the sequence is the set of whole numbers.

The following table shows the pattern.

Month 1 2 3 4 5 6 7 8 9 10

Branches 3 1 2 3 5 8 13 21 34 55Each term is the sum of the previous two terms. This pattern can be written as the recursive formula a0 = 1 , a 1 = 1, a n = a n_ 2 + a n _ x, where n > 2.

f GuidedPractice3. NATURE How many branches will a plant like the one described in Example 3 have after

15 months if no branches are removed?

In Lesson 1-3, you examined the end behavior of the graphs of functions. You learned that as the domains of some functions approach oo, the ranges approach a unique number called a limit. As a function, an infinite sequence may also have a limit. If a sequence has a limit such that the terms approach a unique number, then it is said to converge. If not, the sequence is said to diverge.

V

TechnologyTipConvergent or Divergent Sequences If an explicit formula for a sequence is known, you can enter the formula in the Y = menu of a graphing calculator and graph the related function. Analyzing the end behavior of the graph can help you to determine whether the sequence is convergent or divergent.

B j E u E Convergent and Divergent SequencesDetermine whether each sequence is convergent or divergent.

a. a = 3n + 12The first eight terms of this sequence are 12 ,9 , 6, 3,0, 3, 6, and 9. From the graph at the right, you can see that a does not approach a finite number. Therefore, this sequence is divergent.

8a,

4

O 10 n

b. (i] 36, ua j/ ^ 2The first eight terms of this sequence are 36, 18, 9, -4 .5 ,2 .2 5 , -1 .1 2 5 , 0.5625, -0 .28125 , and 0.140625. From the graph at the right, you can see that a n approaches 0 as n increases. This sequence has a limit and is therefore convergent.

( - 1 ) " n

a,l~ 4h + 1The first twelve terms of this sequence are given or approximated below.

36 a i

>4

12

O 4 10 n

- 12

a1 = 0.2 a 2 = 0.222

3 ~ -0 .231 nA = 0.235

a 5 ~ -0 .2 3 8 a6 = 0.24

a 7 ~ -0 .241 8 0.242

a 9 ~ -0 .2 4 3 a w ~ 0.244

a u ~ -0 .2 4 4 j2 ~ 0.245

3r

O 10 n

-0.

It appears that when n is odd, an approaches j , and when n is even, a n approaches -j. Since a n does not approach one particular value, the sequence has no limit. Therefore, the sequence is divergent.

GuidedPractice

4A- = S 4B. flj = 9, a n = a n _ j + 4 4C. a = 3(1)"

592 Lesson 10-1 Sequences, Series, and Sigma Notation

2 Series A series is the indicated sum of all of the terms of a sequence. Like sequences, series can be finite or infinite. A finite series is the indicated sum of all the terms of a finite

sequence, and an infinite series is the indicated sum of all the terms of an infinite sequence.

StudyTipConverging Infinite Sequences While it is necessary for an infinite sequence to converge to 0 in order for the corresponding infinite series to have a sum, it is not sufficient. Some infinite sequences converge to 0 and the corresponding infinite series still do not have sums.

Sequence Series

Finite 1 ,3 ,5 ,7 ,9 1 + 3 + 5 + 7 + 9

Infinite 1 , 3 , 5 , 7 , 9 , . . . 1 + 3 + 5 + 7 + 9 + . . .

The sum of the first n terms of a series is called the th partial sum and is denoted S. The nth partial sum of any series can be found by calculating each term up to the nth term and then finding the sum of those terms.

The nth Partial Suma. Find the fourth partial sum of a n = (2)" + 3.

Find the first four terms.

fl1 = (2)1 + 3 or 1 /7 = 1

a2 = (2)2 + 3 or 7 n= 2

a 3 = (2)3 + 3 or 5 n = 3

fl4 = (2)4 + 3 or 19 n = 4

The fourth partial sum is S4 = 1 + 7 + (5) + 19 or 22.

b. Find S 3 of a =

Find the first three terms.

a , = - or 0.4 1 101

a , = - or 0.04 2 102

a, = or 0.004 3 103

n= 1

n = 2

n = 3

The third partial sum is S 3 = 0.4 + 0.04 + 0.004 or 0.444.

p GuidedPractice5A. Find the sixth partial sum of a 1 = 8, a n = 0.5(an _ j) , n > 2.

5B. Find the seventh partial sum of a n = 3|^jj

\ Since an infinite series does not have a finite number of terms, you might assume that an infinite series has no sum S. This is true for the series below.

Infinite Sequence

1 ,4 ,7 ,1 0 , . . .

Infinite Series

1 + 4 + 7 + 10 + ..

Sequence of First Four Partial Sums

1, 5 ,12 , 2 2 , . . .

However, some infinite series do have sums. For an infinite series to have a fixed sum S, the infinite sequence associated with this series must converge to 0. Notice the sequence of partial sums in the

infinite series below appears to approach a sum of 0.1 or i .

Infinite Sequence

0 .1 ,0 .0 1 ,0 .0 0 1 ,...

Infinite Series

0 .1 + 0 .0 1 + 0 .001 + ...

Sequence of First Three Partial Sums

0 .1 ,0 .11 ,0 .111 ,...

We will take a closer look at sums of infinite sequences in Lesson 10-3.

< 5

con n ectE D .m cg raw -h ill.co m 1 593

Series are often more conveniently notated using the uppercase Greek letter sigma E . A series written using this letter is said to be expressed using summation notation or sigma notation.

KeyConcept Sigma Notation

ReadingMathSigma Notation J ]a is r e a d \

n=1 /the summation from n = 1 to k of a sub n.

For any sequence av a2, a3, a4 the sum of the first /(term s is denoted

kE an a1 + a2 + a 3 + " + ak>n=1

where n is the index of summation, k is the upper bound of summation, and 1 is the lower bound of summation.

In this notation, the lower bound indicates where to begin summing the terms of the sequence and the upper bound indicates where to end the sum. If the upper bound is given as oo, the sigma notation represents an infinite series.

OO$ > = f l 1 + f l 2 + 3 +=1

B E H uH S D Sums in Sigma Notation

WatchOut!Variations in Sigma Notation The index of summation does not have to be the letter n. It can be represented by any variable. For example, the summation in Example 6a could also be written as

E ( 4 / - 3 ) ./=1

Find each sum.

5a. 5 ] (4- 3 )

n = 1

5

E (4n - 3) = [4(1) - 3] + [4(2) - 3] + [4(3) - 3] + [4(4) - 3] + [4(5) - 3]n=1

= 1 + 5 + 9 + 13 + 17 or 45

b- E 6 - 3n = 3

7

En =36n - 3 6(3) - 3 , 6(4) - 3 , 6(5) - 3 , 6(6) - 3 , 6(7) - 3 + + +

: 7.5 + 10.5 + 13.5 + 16.5 + 19.5 or 67.5

c.1 1 = 1

7 7 7 7 7 7> |- u - I- |- u 10" 101 102 103 104 105

= 0.7 + 0.07 + 0.007 + 0.0007 + 0.00007 + ...

= 0.77777... or ^

"p GuidedPractice 6A. E -

13

n= 16B. E ( 3 - " 2)

n = 76C- E ^

n = l 1U

Note that while the lower bound of a summation is often 1, a sum can start with any term p in a sequence as long as p < k. In Example 6b, the summation started with the 3rd term of the sequence and ended with the 7th term.

594 | Lesson 10-1 j Sequences, Series, and Sigma Notation

Exercises = Step-by-Step Solutions begin on page R29.

Find the next four terms of each sequence. (Example 1)

1. 1 ,8 ,1 5 ,2 2 , . . . 2. 3 , - 6 , 1 2 , - 2 4 , . . .

3. 8 1 ,2 7 ,9 ,3 , . . . 4. 1 ,3 ,7 ,1 3 , . . .

5. - 2 , - 1 5 , - 2 8 , - 4 1 . . . 6. 1 ,4 ,1 0 ,1 9 , . . .

Find the first four terms of each sequence. (Example 1)

7. an = n2 1

8. an = - 2 n + 7

10. an = ( - 1 ) + 1 + n

11. AUTOMOBILE LEASES Lease agreements often contain clauses that limit the number of miles driven per year by charging a per-mile fee over that limit. For the car shown below, the lease requires that the number of miles driven each year must be no more than 15,000. (Example 2)

a. Write the sequence describing the maximum number of allowed miles on the car at the end of every 12 months of the lease if the car has 1350 miles at the beginning of the lease.

b. Write the first 4 terms of the sequence that gives the cumulative cost of the lease for a given month.

c. Write an explicit formula to represent the sequence in part b.

d. Determine the total amount of money paid by the end of the lease.

Find the specified term of each sequence. (Example 2)

12. 4th term, Aj = 5,% = 3an _ 1 + 10, n > 2

13. 7th term, a 1 = 14, an = 0.5an _ 1 + 3, n > 2

14. 4th term, = 0, an = 3fl - \ n > 2

15. 3rd term, a 1 = 3, a n = ( _ j ) 2 5an _ 1 + 4, n > 2

16. WEBSITE Khari, the student from the beginning of the lesson, had great success expanding her Web site. Each student who received a referral developed a Web page and referred five more students to Khari's site. (Example 3)

a. List the first five terms of a sequence modeling the number of new Web pages created through Khari's site.

b. Suppose the school has 1576 students. After how many rounds of referrals did the entire student body have a Web page?

( l 7 | BEES Female honeybees come from fertilized eggs (male and female parent), while male honeybees come from unfertilized eggs (one female parent). (Example 3)

a. Draw a family tree showing the 3 previous generations of a male honeybee (parents only).

b. Determine the number of parent bees in the 11th previous generation of a male honeybee.

Determine whether each sequence is convergent or divergent. (Example 4)

18. a 1 = 4 , 1.5an _ j , n > 2

20. a n = n2 8n + 106

22. 2 = 1, ciyi 4 _

n > 2n2 + 4

19. a =10"

21. a 1 64, a n _ v n > 2

23. an = n2 3n + 1

24. a

26. a n =

n 13 + n 5n + 6

25. a j = 9 , a n = - 2

5n

+ 3

27. a5n + 1

Find the indicated sum for each sequence. (Example 5)

28. 5th partial sum of an = n(n 4)(w 3) _L 'X

29. 6th partial sum of a = -----------

30. S 8 of flj = 1, a n = a n _ 1 + (18 n ) ,n > 2

31. S 4 of a 1 = 64, an = a n _ v n > 2

32. 11th partial sum of a 1 = 4 , a n = (I) '1 a n _ 1 1 + 3), n > 2

33. S 9 of flj = 35, a n = a n _ x + 8, n > 2

34. 4th partial

35. of

34. 4th partial sum of a x = 3, an = 2)3, n > 2(-3 )"

10

Find each sum. (Example 6)\ 8

,36.,, Y , (6n - 11)n = 1

O 7

37. X ] (30 - 4m)n = 4

I 3 8 . / E [ > - 5 )] n = l

M E ( f - 7 )

39. ] > ] ( n 2 - 6 n + l )m= 2

10

27 E [ ( -2 ) " - 9]n 0

41. Y K - 4 ) 2( - 5)]n=l

43

44.OO . .

S5(w) 45. En = l 10"46. FINANCIAL LITERACY Jim 's bank account had an initial

deposit of $380, earning 3.5% interest per year compounded annually.

a. Find the balance each year for the first five years.b. Write a recursive and an explicit formula defining his

account balance.

C. For very large values of n, which formula gives a more accurate balance? Explain.

&

connectED.mcgraw-hill.com 1 595

47. INVESTING Melissa invests $200 every 3 months. The investment pays an annual percentage rate of 8%, and the interest is compounded quarterly. If Melissa makes each payment at the beginning of the quarter and the interest is posted at the end of the quarter, what will the total value of the investment be after 2 years?

48. RIDES The table shows the number of riders of the Mean Streak each year from 1998 to 2007. This ridership data

can be approximated by a n = -^n + 1.3, where n = 1

represents 1998, n = 2 represents 1999, and so on.

Mean Streak Roller Coaster

YearNumber of

Riders (millions)

YearNumber of

Riders (millions)

1998 1.31 2003 0.99

1999 1.15 2004 0.95

2000 1.14 2005 0.89

2001 1.09 2006 0,81

2002 1.05 2007 0.82

Source: Cedar Fair Entertainment Company

a. Sketch a graph of the number of riders from 1998 to 2007. Then determine whether the sequence appears to be convergent or divergent. Does this make sense in the context of the situation? Explain your reasoning.

b. Use the table to find the total number of riders from 1998 to 2005. Then use the explicit sequence to find the 8th partial sum of a n. Compare the results.

C. If the sequence continues, find a u . What does this number represent?

Copy and complete the table.

Recursive Formula Sequence ExplicitFormula

49. 6 ,8 ,1 0 ,1 2 , . . .

50. ai = 15. a = a _ 1 - 1 , / ? > 2

51. 7 ,2 1 ,6 3 ,1 8 9 , ...

52. a = 1 0 ( - 2 ) n

53. a = 8/? - 3

54. a1 = 2 ,a = 4 a _ 1, n > 2

55. a, = 3, a = an _ , + 2 / ? - 1, n> 2

56. a = n2 + 1

Write each series in sigma notation. The lower bound is given.

57. - 2 - 1 + 0 + 1 + 2 + 3 + 4 + 5; n = 1

C O 1 . 1 . 1 . 1 . 1 . 1 . __ A 20 25 30 35 40 45'

@ ( 8 + 27 + 64 + + 1000; n = 2

60' 2 + + i + h 128; " = 1

61. - 8 + 16 - 32 + 64 - 128 + 256 - 512; n = 3

E 8H ) +8 ( l ) +8 ( ^ ) + ' " +8( - # =1

596 | Lesson 10-1 | Sequences, Series, and Sigm a N o ta tio n

D eterm ine w hether each sequence is convergent or divergent. Then find the fifth partial sum of the sequence.

63. a = s in J 64. an = n cos tt 65. a n = e 2 cos im

66. WATER PRESSURE The pressure exerted on the human body at sea level is 14.7 pounds per square inch (psi). For each additional foot below sea level, the pressure is about 0.445 psi greater, as shown.

sea level = 14.7 psi

" x + 0 .4 4 5 psi/ftN

o *0> N

a. Write a recursive formula to represent a n, the pressure at n feet below sea level. (Hint: Let a 0 = 14.7.)

b. Write the first three terms of the sequence and describe what they represent.

C. Scuba divers cannot safely dive deeper than 100 feet. Write an explicit formula to represent an. Then use the formula to find the water pressure at 100 feet below sea level.

M atch each sequence w ith its graph.

68. [16

a ,

12

8

4

O 2 n

69.

71.

a. fl = 3

c- = ( - ! )

e. a = 9 2n

8a

4

O . in

- 4

- 8

a

8

4

O 2 in

- 4

24a ,

18

12

-6

O 4 6 8 h

70.

72.

b- an = ~ j n + 9

d. a n

f. a ( ! ) '

16a r

12

8

4

O 4 n

a ,

8 i

O 2 n

- 8

16

f(2 M)

+ 8

73. GOLDEN RATIO Consider the Fibonacci sequence 1 ,1 ,2,3, 2 + a-i-

a. Find a - for the second through eleventh terms ofn 1

the Fibonacci sequence.

b. Sketch a graph of the terms found in part a. Let n 1 be the x-coordinate and a" be the y-coordinate.

an - 1 J

C. Based on the graph found in part b, does this sequence appear to be convergent? If so, describe the limit to three decimal places. If not, explain why not.

d. In a golden rectangle, the ratio of the length to the width is about 1.61803399. This is called the golden ratio. How

does the limit of the sequence ^ n compare to the golden ratio?

e. Golden rectangles are common in art and architecture. The Parthenon, in Greece, is an example of how golden rectangles are used in architecture.

Research golden rectangles and find two more examples of golden rectangles in art or architecture.

Determine whether each sequence is convergent or divergent.

74. 7 ja t 75i16

3r

8

O n

- 8

16

OCNJ...T..

O - 12 n

BO

! 1

76.^4

3 n

O ! * 1 ? *16

- 4

- 8I

I12

1

77.4 an

-3

2

-1n

O 4 12 16

Write an explicit formula for each recursively defined sequence.

78. a x = 10; a n = a n _ j + 5

79. f l j = 1.25; a n = a n _ 1 0.5

80. f l j = 128; an = 0.5a n _ x

81. 0 MULTIPLE REPRESENTATIONS In this problem, you will investigate sums of infinite series.

a. NUMERICAL Calculate the first five terms of the infinite

sequence a n =

b. GRAPHICAL Use a graphing calculator to sketch

C. VERBAL Describe what is happening to the terms of the sequence as n > oo.

d. NUMERICAL Find the sum of the first 5 terms, 7 terms, and 9 terms of the series.

e. VERBAL Describe what is happening to the partial sums S n as n increases.

f. VERBAL Predict the sum of the first n terms of the series. Explain your reasoning.

H.O.T. Problems Use Higher-Order Thinking Skills82. CHALLENGE Consider the recursive sequence below.

- n_2fo r fli = i , a2 = 1, n > 3

a. Find the first eight terms of the sequence.

b. Describe the similarities and differences between this sequence and the other recursive sequences in this lesson.

83. OPEN ENDED Write a sequence either recursively or explicitly that has the following characteristics.

a. converges to 0

b. converges to 3

c. diverges

84. WRITING IN MATH Describe why an infinite sequence must not only converge, but converge to 0, in order for there to be a sum.

REASONING D e te r m in e w h e th e r e a c h s ta te m e n t is true or f a ls e . E x p la in y o u r re a s o n in g .

5 5 5

85. E ( 2 + 3 ) = X / nl + nn = 1 n = 1 n = 1

5 7

86. E 3" = E 3 2n = 1 n = 3

(8 7 | CHALLENGE Find the sum of the first 60 terms of the sequence below. Explain how you determined your answer.

1 5 ,1 7 ,2 , - 1 5 , - 1 7 , .. . , where an = an ] an _ 2 for n > 3

88. WRITING IN MATH Make an outline that could be used to describe the steps involved in finding the 300th partial sum of the infinite sequence an = 2n 3. Then explain how to express the same sum using sigma notation.

connectED.mcgraw-hill.com I 597

Spiral Review

Graph each complex number on a polar grid. Then express it in rectangular form. (Lesson 9-5)

89,, 2 ( 5ir . . 5tt\ cos + i sin 4 4 / 90. 2.5 (cos 1 + i sin 1) 91. 5(cos 0 + i sin 0)

Determine the eccentricity, type of conic, and equation of the directrix given by each polar equation. (Lesson 9-4)

3 no .. _ 6 m 192. r =2 0.5 cos i

93. r 1.2 sin 9 + 0.3 94. r = 0.2 - 0.2 sin 9

Determine whether the points are collinear. Write yes or no. (Lesson 8-5)

95. ( - 3 , - 1 ,4 ) , (3, 8 ,1 ), (5 ,12, 0) 96. (4, 8, 6), (0, 6 ,12 ), (8 ,10 , 0)

97. (0, - 4 ,3 ) , (8, - 1 0 , 5), (12, - 1 3 , 2) 98. ( - 7 , 2, - 1 ) , ( - 9 ,3 , - 4 ) , ( - 5 ,1 , 2)

Find the length and the midpoint of the segment with the given endpoints. (Lesson 8-4)

99. (2, -1 5 ,1 2 ) , (1, -1 1 ,1 5 ) 100. ( - 4 ,2 ,8 ) , (9, 6, 0) 101. (7 ,1 , 5), ( - 2 , - 5 , -1 1 )

102. TIMING The path traced by the tip of the hour-hand of a clock can be modeled by a circlewith parametric equations x = 6 sin t and y = 6 cos t. (Lesson 7-5)

a. Find an interval for t in radians that can be used to describe the motion of the tip as it moves from 12 o'clock noon to 12 o'clock noon the next day.

b. Simulate the motion described in part a by graphing the equation in parametric mode on a graphing calculator.

C. Write an equation in rectangular form that models the motion of the hour-hand.Find the radius of the circle traced out by the hour-hand if z and y are given in inches.

Find the exact value of each expression. (Lesson 5-4)

103. tan12

104. sin 75 105. cos 165

Find the partial fraction decomposition of each rational expression. (Lesson 6 -

10x2 l lx + 4

You found terms of sequences and sums of series.(Lesson 10-1)

* 1

2

Find nth terms and arithmetic means of arithmetic sequences.

Find sums of n terms of arithmetic series.

With cross country season approaching, Meg decides to train every day until the first day of practice. She plans to run 1 mile the first day, 1.25 miles the second day, 1.5 miles the third day, and so on. Her goal is to run a total of 100 miles before the first day of practice.

NewVocabularyarithmetic sequence common difference arithmetic means first difference second difference arithmetic series

1 Arithmetic Sequences A sequence in which the difference between successive terms is a

constant is called an arithmetic sequence. The constant is referred to as the common difference, denoted d. To find the common difference of an arithmetic sequence, subtract any term

from its succeeding term. To find the next term in the sequence, add the common difference to the given term.

Arithmetic SequencesDetermine the common difference, and find the next four terms of the arithmetic sequence 17 ,12 , 7 , . . . .

First, find the common difference.

02 dj = 12 17 or 5 Find the difference between tw o pairs of consecutiveterms to verify the common difference,

a 02 = 7 12 or 5

The common difference is 5. Add 5 to the third term to find the fourth term, and so on.

a4 = 7 + ( - 5 ) or 2 a5 = 2 + ( - 5 ) or - 3 a6 = - 3 + ( - 5 ) or - 8 a7 = - 8 + ( - 5 ) or - 1 3

The next four terms are 2, 3, 8, and 13.

! GuidedPracticeDetermine the common difference, and find the next four terms of each arithmetic sequence.

1A. - 1 2 9 , - 9 8 , - 6 7 , . . . 1B. 2 4 4 ,1 8 7 ,1 3 0 ,...

Each term in an arithmetic sequence is found by adding the common difference to the preceding term. Therefore, an = an_ 1 + d. You can use this recursive formula to develop an explicit formula for generating an arithmetic sequence. Consider the arithmetic sequence in which flj = 6 and d = 3.

first term a i 6

second term Cl2 o d 6 + 1(3) = 9

third term a3 ci t- 2 d 6 + 2(3) = 12

fourth term fl4 ox -t- 3 d 6 + 3(3) = 15fifth term a5 o l - 4tf 6 + 4(3) = 18

nth term an flj + (n 1 )d 6 + (n - 1)3

The pattern formed leads to the following formula for finding the nth term of an arithmetic sequence.

Ke Concept The nth Term of an Arithmetic SequenceWords The nth term of an arithmetic sequence with first term a1 and common difference d is

given by a = a 1 + (n - 1 )d.

Example The 16th term of 2, 5, 8, . . . is a 16 = 2 + (16 - 1) 3 or 47.

599

StudyTipExplicit Formulas If a term other than a, is given, the explicit formula for finding the nth term of a sequence needs to be adjusted. This can be done by subtracting the number of the term given from n. For example, if a5 is given, the equation would become a = a5 + (n - 5 )d, or if a0 is given, then 3/7 = a0 + nd.

H E Z 5 2 J E E Explicit and Recursive FormulasFind both an explicit formula and a recursive formula for the nth term of the arithmetic sequence 12, 2 1 , 3 0 , . . . .

First, find the common difference.

Find the difference between two pairs of consecutive terms to verify the common difference.

u2 flj = 21 12 or 9

a3 2 = 30 21 or 9

For an explicit formula, substitute al = 12 and d arithmetic sequence.

> in the formula for the nth term of an

an = a 1 + (n 1 )d

= 12 + (n - 1)9

= 12 + 9 (n 1) or 9n 4- 3

nth term of an arithmetic sequence

a , = 12 and d = 9

Simplify.

For a recursive formula, state the first term a 1 and then indicate that the next term is the sum of the previous term an _ x and d.

a l = 12, a n = an _ l + 9

GuidedPractice2. Find both an explicit formula and a recursive formula for the nth term of the arithmetic

sequence 3 5 ,2 3 ,1 1 , . . . .

The formula for the nth term of an arithmetic sequence can be used to find a specific term in a sequence.

StudyTipRate of Change Arithmetic sequences have a constant rate of change which is equivalent to the common difference d.

S E E ] "th TermsFind the 68th term of the arithmetic sequence 25 ,17 , 9, . . . .

First, find the common difference.

Find the difference between two pairs of consecutivea2 = 17 25 or

a3 a2 = 9 17 or 8terms to verify the common difference.

Use the formula for the nth term of an arithmetic sequence to find a6

an = a l + (n 1 )d nth term of an arithmetic sequence

6 8 = 25 + ( 6 8 - 1)(8 )

a6 8 = 511

n = 68, a , = 25, and d = 8

Simplify.

b. Find the first term of the arithmetic sequence for which a25 = 139 and d = .Substitute a 2 5 = 139, n = 25, and d = in the formula for the nth term of an arithmetic sequence to find a v

nth term of an arithmetic sequencea,, = al + ( 1 )d

139 = flj + (25 - 1)|

ax = 1 2 1

n = 2 5 ,a = 1 3 9 , a n d rf = |

Simplify.

y GuidedPractice3A. Find the 38th term of the arithmetic sequence 29, 2, 25, .. . .

3B. Find d of the arithmetic sequence for which ax = 75 and a38 = 56.5.

If two nonconsecutive terms of an arithmetic sequence are known, the terms between them can be calculated. These terms are called arithmetic means. In the sequence below, 17,27, and 37 are the arithmetic means between 7 and 47.

- 3 , 7 ,17, 27, 37, 47, 57

600 | Lesson 10-2 | A rith m e tic Sequences and Series

StudyTipAlternative Method An alternative method to find (/would be to subtract the first term from the last term and divide by the totalnumber of terms minus 1.

V ................................ J

Arithmetic MeansW rite an arith m etic seq u en ce th at h as fo u r arith m etic m ean s b e tw een 4.3 and 12.8.

T he sequence w ill resem ble 4.3, .? . . , .? , 12.8. N ote that 12.8 is the sixth term of thesequence or a6.

First, find the com m on d ifference using a6 = 12.8, flj = 4.3, and n = 6 .

an = dy + (n 1 )d nth term of an arithm etic sequence

12.8 = 4.3 + (6 - 1 )d a = 12.8, a, = 4.3, and n = 6

12.8 = 4.3 + 5d Simplify.

d = 1 .7 Solve for of.

Then determ ine the arithm etic m eans b y using d = 1.7.

a 2 = 4.3 + 1.7 or 6

a 3 = 6 + 1.7 or 7.7

4 = 7.7 + 1.7 or 9.4

a5 = 9.4 + 1.7 or 11.1

The sequence is 4 .3, 6, 7 .7 ,9 .4 ,1 1 .1 ,1 2 .8 .

^ GuidedPractice4. W rite a sequence that has six arithm etic m eans betw een 12.4 and 24.7.

The firs t d iffe ren ces of a sequence are found b y subtracting each term from its successive term .

Sequence Cly #2/ #3, #4, #5,

1st differences d d d d

W hen the first differences are all the sam e, the sequence is arithm etic and the nth term can be m odeled b y a linear function o f the form a n = dn + a0, as show n.

If the first differences are not the sam e, the sequence is not arithm etic. H ow ever, the differences m ay still help to identify the type of function that can be used to m od el the sequence. C onsecutive first differences m ay be subtracted from one another, thus prod ucing second d ifferen ces.

Sequence

1st differences

2nd differences

12, 20, 30, 42, 5 6 , . . .

10 12 14

If the second differences are constant, then the nth term o f the sequence can be m odeled by a quadratic function. This function can be found b y solv ing a system o f equations, as dem onstrated in Exam p le 5.

connectED. m cgraw -h i i lx o m ^ 6 0 1

S 2 2 2 Q 3 0 Second Differences

Bn n + 5n + 6

an 140

20

6 2 n

Figure 10.2.1

Find a quad ratic m od el fo r the seq u en ce 12, 20, 30, 42, 56, 72, . . . .

The th term can be represented b y a quadratic equation of the form an = an2 + bn + c. Substitute values for a and n into the equation.

a = 12 and /? = 1

: 20 and n = 2

: 30 and n = 3

12 = M l)2 + M l) + C

20 = M2)2 + M2) + c

30 = M3)2 + M3) + c

This yields a system o f linear equations in three variables.

12 = a + b + c Simplified firs t equation

20 = 4a + 2b + c Simplified second equation

30 = 9a + 3b + c Simplified th ird equation

Solving for a, b, and c gives a = 1, b = 5, and c = 6 . Substitu ting these values in the equation for an, the m odel for the sequence is a,, = n2 + 5n + 6, as show n in Figure 10.2.1.

p GuidedPractice5. Find a quadratic m odel for the sequence 14, 8, 0 ,1 0 , 22, 36, . . . .

StudyTipHigher Differences The number of variables in the standard form of the equation dictates the number of equations needed in the system formed.

If calculating second differences does not result in a constant ^difference, h igher differences m ay be found. This process is sim ilar to the process needed for finding a quadratic equation. The function that w ill m odel a sequence is d ependent up on how m any com puted differences are necessary before finding a constant d ifference.

H igher differences m ay never result in constant differences. In this case, there m ay not be a polynom ial m odel that can be used to describe the sequence.

Differences Model

first linear

second quadratic

third cubic

fourth quartic

fifth quintic

2 Arithmetic Series A n arith m etic series is the indicated sum of the term s o f an arithm etic sequence.

A rithm etic S eq u en ce A rith m etic Series

- 6, - 3 , 0 , 3 , 6 - 6 + ( - 3 ) + 0 + 3 + 6

4 .2 5 ,4 , 3.75, 3.5, 3.25 4.25 + 4 + 3.75 + 3.5 + 3.25

ay Oj, fl3, a4, an ax + ^ + a + a + + an

To develop a form ula for finding the sum of a fin ite arithm etic series, start by looking at the series S that has term s created by adding m ultiples o f d to av If w e com bine this w ith the sam e series w ritten in reverse order, w e can find a form ula for calcu lating the sum of a finite arithm etic series.

Sn = + (flj + d) + (flj + 2d) + + (an 2d) + (afl d) + an

(+ ) Sn = On + (all d) + (an 2d) + + (ax + 2d) + (ax + d) + a

2 Sn = (a1 + an) + (fl] + a^ + (a + an) + + (a + an) + (ax + an) + (tt [ + a )

2Sn = n(a1 + an) There are n terms in the series, all of which are (a, + an).

Therefore, Sn = ^(al + a j . W hen the value of the last term is unknow n, you can still determ ine the

wth partial sum o f the series by com bining the ?7th term of an arithm etic sequence form ula and the sum of a finite arithm etic series form ula.

Sn = ( f l j + a) Sum of a fin ite arithm etic series formula

Sn = + [d j + (n 1 )d ]} a= + (n 1 )d, /7th term of an arithm etic sequence formula

Sn = y [2 flj + (n 1 )d] Simplify.

6 0 2 | Lesson 10-2 | Arithm etic Sequences and Series

StudyTipArithmetic Series All infinite arithmetic sequences diverge except for those in which d = 0. As a result, only a finite arithmetic series or the nth partial sum of an infinite arithmetic series can be calculated.

KeyConcept Sum of a Finite Arithmetic Series

The sum of a finite arithmetic series with n terms or the nth partial sum of an arithmetic series can be found using one of two related formulas.

Formula 1

Formula 2

S/j - ? ( a i + an)

Sn = j [ 2 a , + ( n - m

Sum of Arithmetic Series

Fin d the ind icated sum o f each arith m etic series,

a. 5 + 2 + 9 H h 317

In this sequence, = 5, a,, = 317, and d = 2 (5) or 7. U se the nth term form ula to find the num ber o f term s in the sequence n.

an = flj + (n 1 )d nth term of an arithm etic sequence

317 = 5 + (n 1)7 a = 317, a, = 5, and cf= 7

4 7 = n Simplify.

N ow use Form ula 1 to find the sum o f the series.

S = y ( i + )

S 47 = ^ f ( - 5 + 3 1 7 )

2 V

4 7 ,2

: 7332

Formula 1

n = 47, a , :

Simplify.

-5 , and a = 317

b . th e 28th partia l sum o f 27 + 1 4 + 1 +

In this sequence, flj = 5 and d = 14 27 or 13. U se Form ula 2 to find the 28th partial sum.

S = j[2 n1 + (n l)d ]

S 28 = ^ f [ 2 ( 2 7 ) + ( 2 8 - 1 ) ( 1 3 ) ]

-4158

Formula 2

n = 28, a1 = 27, and d-

Simplify.

-13

28

C. E ( 5 m 1 7 )M=6

2 8

E (5m - 17) = [5(6) - 17] + [5(7) - 17] + + [5(28) - 17]n = 6

= 13 + 18 + + 123

The first term o f this series is 13 and the last term is 123. T he num ber of term s is equal to the upper bound m inus the low er bound plus one, w hich is 28 6 + 1 or 23. Therefore ax = 13, an = 123, and n = 23. U se Form ula 1 to find the sum of the series.

So + )

(1 3 + 1 2 3 )

2

2 3 ,23 2

= 1564

GuidedPractice

Formula 1

n = 23, a1 = 13, and a = 123

Simplify.

6A. 211 + 193 + 175 + + ( -4 5 5 )37

6C. Y j i 2 + 3)n = 2 3

6B. the 19th partial sum of 19 + 23 + 65 +18

6D. E ( - 2w + 5 7 )n= 12

^|[conmec ] 603

Real-WoridCareerSoftware Engineer Most video game programmers are software engineers who plan and write game software. Most programmers have a bachelors degree in computer science, information systems, or mathematics. Some also obtain technical or professional certification.

A rithm etic series have m any useful real-life applications.

Real-World Example 7 Sum of an Arithmetic SeriesVIDEO GAMES A v id eo gam e tournam ent, in w h ich gam ers com pete in m u ltip le gam es and accum ulate an ov erall score, pays th e top 20 fin ish e rs . F irst p lace receives $5000, second place receives $4800, th ird p lace receiv es $4600, and so on. H ow m u ch to ta l p rize m o n ey is aw arded?

S = [2 a , + (h - 1 )d]

s 20 = ^ [2 (5 0 0 0 ) + (20 -

= 62,000

1)(200)]

Formula 2

n = 20, a, = 5000, and d = - 2 0 0

Simplify.

The total prize m oney aw arded is $62,000

GuidedPractice7. VIDEO GAMES Selm a is p laying a video gam e. She scores 50 points if she clears the first level.

Each follow ing level is w orth 50 m ore points than the previous level. Thus, she scores 100 points for clearing the second level, 150 for the third , and so on. W hat is the total am ount of p oints Selm a w ill score after she clears the n inth level?

The form ula for the sum of a finite arithm etic series can also b e used to solve for values of n.

BASEBALL C arter has b e e n co llectin g b a seb a ll cards sin ce h is fa th er gave h im a 20-card co llectio n . D u rin g each m on th , C arter's fa th er g ives h im 5 m ore cards than the previous m onth . In how m any m on ths w ill C arter reach 1000 cards?

S = | [ 2 a , + ( n - 1 )d] Formula 2

1 0 0 0 = | [ 2 ( 2 0 ) + (n - 1 )5 ] S = 1000, a , = 2 0 , and d = 5

2 0 0 0 = n(5n + 3 5 ) Multiply each side by 2 and simplify.

0 = 5 n2 + 3 5 n - 2 0 0 0 Distribute and subtract 2000 from each side.

0 = n2 + 7n 4 0 0 Divide each side by 5.

- 7 V ? 2 - 4 (1 )(400)Use the Quadratic Formula.

" ~ 2(1)

n ~ 1 6 .8 and 2 3 .8 Simplify,

Because tim e cannot be negative, C arter w ill reach 1 0 0 0 cards in 1 7 m onths.

CHECK 2 0 + 2 5 + + 100 = (2 0 + 1 0 0 )

= 1 0 2 0

In seventeen m onths, C arter w ill have 1 0 2 0 baseball cards, w hich is m ore than 1000 .

f GuidedPractice8 . LAWN SERVICE K evin runs a law n m ow ing service. H e currently has 14 clients. H e has gained

2 new clients at the beginning o f each of the past three years. E ach year, he m ow s each client's law n an average of 15 tim es. Starting now, if K evin continues to gain 2 clients each year and if he charges $30 per law n, after how m any years w ill he earn a total of $51,300?

6 0 4 I Lesson 10-2 j Arithm etic Sequences and Series

Exercises = Step-by-Step Solutions begin on page R29.

D eterm ine the com m on d ifferen ce , and fin d the next fo u r term s o f each arithm etic sequ ence. (Example 1)

1. 2 0 ,1 7 ,1 4 , . . .

3. 1 1 7 ,1 0 8 ,9 9 , . . .

5. - 3 , 1 , 5 , . . .

7. - 4 . 5 , - 9 . 5 , - 1 4 . 5 , . . .

2. 3 ,1 6 ,2 9 , . . .

4. - 8 3 , - 6 1 , - 3 9 , . . .

6 . 4 ,2 1 ,3 8 , . . .

8 . - 9 7 , - 2 9 , 3 9 , . . .

9'J MARCHING BAND A m arching band begins its perform ance in a pyram id form ation. The first row has 1 band m em ber, the second row has 3 band m em bers, the third row has 5 band m em bers, and so on. (Examples 1 and 2)

a. Find the num ber of band m em bers in the 8th row.

b. W rite an explicit form ula and a recursive form ula for finding the num ber of band m em bers in the nth row.

Find bo th an exp licit form ula and a recursive form u la for the wth term o f each arith m etic seq u en ce. (Example 2)

10. 2 , 5 , 8 , . . .

12. - 9 , - 1 6 , - 2 3 , . . .

14. 2 5 , 1 1 , - 3 , . . .

16. - 1 8 , 4 , 2 6 , . . .

11. - 6 , 5 , 1 6 , . . .

13. 4 ,1 9 ,3 4 , . . .

15. 7 , - 3 . 5 , - 1 4 , . . .

17. 1 ,3 7 ,7 3 , . . .

Find the sp ecified valu e fo r the arith m etic seq u en ce w ith the g iven characteristics. [Example 3)

18. If au = 85 and d = 9 , find av

19. Find d for 24, 31, 3 8 , . . . .

20. If an = 14, flj = 36, and d = 5, find n.

21 . If flj = 47 and d = 5, find a12.

22. If a , , = 95 and a1 = 11, find d.

23. Find a6 for 84, 5, - 7 4 , . . . .

24. If an = 20, flj = 46, and d = 11, find n.

25. If a35 = 63 and ax = 39, find d.

26. CONSTRUCTION Each 8-foot section of a w ooden fence contains 14 pickets. L et an represent the num ber o f pickets in n sections. Example 3)

a. Find the first 5 term s of the sequence.

b. W rite a recursive form ula for the sequence in p art a.C. If 448 pickets w ere used to fence in the cu stom er's

backyard, how m any feet of fencing w as used?

Find the ind icated arithm etic m ean s fo r each set o f nonconsecu tive term s. (Example 4)

27. 3 m eans; 19 and 5 28. 5 m eans; 62 and 8

29. 4 m eans; 3 and 88 30. 8 m eans; 5.5 and 23.75

31 . 7 m eans; 4.5 and 7.5 32. 10 m eans; 6 and 259

F ind a qu ad ratic m od el fo r each seq u en ce. (Example 5)

33. 1 2 ,1 9 ,2 8 ,3 9 ,5 2 ,6 7 , . . .

34. - 1 1 , - 9 , - 5 , 1 , 9 , 1 9 , . . .

35. 8 , 3 , - 6 , - 1 9 , - 3 6 , - 5 7 , . . .

36. - 7 , - 2 , 9 , 2 6 , 4 9 , 7 8 , . . .

37. 6, - 2 , - 1 2 , - 2 4 , - 3 8 , - 5 4 , . . .

38. - 3 ,1 , 1 3 , 3 3 , 6 1 , 9 7 , . . .

F in d the in d ica ted sum o f each arith m etic series. [Example 6)

39. 26th partial sum of 3 + 15 + 27 + + 303

40. - 2 8 + ( - 1 9 ) + ( - 1 0 ) + + 242

41 . 42nd partial sum of 120 + 114 + 108 +

42. 54th partial sum of 213 + 205 + 197 +

43. - 1 7 + 1 + 19 + + 649

44. 89 + 58 + 2 7 + + ( - 5 6 2 )

45. RUNNING Refer to the beginn ing of the lesson. (Example 6)

a. D eterm ine the num ber of m iles M eg w ill run on her 12th day o f training.

b. D uring w hich day o f training w ill M eg reach her goal o f 100 total m iles?

Fin d the in d icated sum o f each arith m etic series. (Example 6)2 0

46. J 2 ( 3 + 2n)n = l

2 8

47. X ( 10 - 4 " )n = 1

18

48. J ] ( - - 9 n - 2 6 )n = 7

5 2

49. X ( 7 + ! )n = 6

4 2

50. E ( 8 4 - 3)n = 7

1 3

51. X [32 + 4(n - 1)]n = 1

2 4

52. ( f - 9 )n=2 0 VZ '

9

53. X ( - 15 - 12)71=2

54. CONSTRUCTION A crew is tiling a hotel lobby w ith a trapezoidal m osaic pattern. The shorter base o f the trapezoid begins w ith a row of 8 tiles. E ach row has two additional tiles until the 20th row. D eterm ine the num ber of tiles needed to create the m osaic design. (Example 7)

55. SN0WM0BILING A snow m obiling com petitor travels 12 feet in the first second of a race. If the com petitor travels 1.5 additional feet each subsequent second, how m any feet did the com petitor travel in 64 seconds?(Example 7)

|l||U Second 1 Second 2 Second 3

- 1 2 f t - -13 .5 f t - 1 5 f t -

mm

&connectED.mcgra^hiTuon 6 0 5

56. FUNDRAISING Lalana organized a charity w alk. In the first year, the w alk generated $3000. She hopes to increase this am ount by $900 each year for the next several years. If her goal is m et, in how m any years w ill the w alk have generated a total of at least $65,000?

57. Find an if S = 490, = 5, and n = 100.

58. If Sn = 51.7, n = 22, a,, = -1 1 .3 , find av

59. Find n for 7 + ( 5.5) + ( 4) + if Sn = 14 and a,, = 3.5.

60. Find if Sn = 1287, n = 22, and d = 5.

61. If S26 = 1456, and = 19, find d.

62. If S12 = 174, a12 = 39, find d.

W rite each arithm etic series in sigm a n otation . T h e low er b ou n d is given.

63. 6 + 12 + 18 + + 66; tt = 1

64. - 1 + 0 + 1 + + 7 ;n = 1

65. 17 + 21 + 25 + + 61; = 4

66. 1 + 0 + ( - 1 ) + ( - 2 ) + + ( - 1 3 ) ; n = 6

e , _ f + ( _ f ) + ( _ a ) + .. . + (_ | ) : = 2

68. 9.25 + 8.5 + 7.75 + ------ 2; w = 1

69. CONCERTS The seating in a concert auditorium is arranged as show n below.

M M N H d t * j j j f Row35

34 seats i t v i t r Row 3 , 29 seats M W W W W Row 2

1 * * * 1 24 seats m r n r n Row 1

* Stage *

a. W rite a series in sigm a notation to represent the num ber of seats in the auditorium , if the seating pattern show n in the first 3 row s continues for each successive row.

b. Find the total num ber of seats in the auditorium .

C. A nother auditorium has 32 row s w ith 18 seats in the first row and 4 m ore seats in each of the successive rows. H ow m any seats are there in this auditorium ?

W rite a fu n ction that can b e used to m od el the nth term o f each sequ ence.

70. 2 ,5 ,8 ,1 1 ,1 4 ,1 7 , . . .

71. 8 ,1 3 ,2 0 ,2 9 ,4 0 ,5 3 , . . .

72. 2 ,2 ,4 ,8 , 1 4 ,2 2 , . . .

73. 5 ,3 1 ,9 7 ,2 2 1 ,4 2 1 , 7 1 5 , . . .

74. - 6 , - 8 , - 6 , 6 , 3 4 , 8 4 , . . .

75. 0, 2 3 ,1 3 4 , 447 ,1124 , 2375, . ..

1 0 0 6 5 .

76. E ( 6 t t + 2) 77. 8 - f -n = 1 n = 2 1 v

778. #22 63, #29 7 79. flg = 4, a ^ j

80. CALCULUS The area betw een the graph of a continuous function and the x-axis can be approxim ated using sequences. C onsider/(x) = x 2 on the interval [1, 3].

a. W rite the sequence x form ed w hen there are 5 arithm etic m eans betw een 1 and 3.

b. W rite the sequence yn form ed w hen yn =/(x).

C. W rite the sequence pn defined by d yn.

d. The left-hand approxim ation of the area is givenn

b Y Ln = YlVk- F in d L 6.k = l

e. The right-hand approxim ation of the area is given byn + 1

R n = Y j P t F i n d K 6.k 2

Find each common difference.

H.O.T. Problems Use Higher-Order Thinking Skills81. ERROR ANALYSIS Peter and C andace are given the

arithm etic sequence 2, 9 , 1 6 , . . . . Peter w rote the explicit form ula an = 2 + 7( 1) for the sequence. C andace's form ula is an = 7n 5. Is either o f them correct? Explain.

82. OPEN ENDED You have learned that the nth term of an arithm etic sequence can be m odeled by a linear function. C an the sequence of partial sum s of an arithm etic series also be m odeled b y a linear function? If yes, provide an exam ple. If no, how can the sequence be m odeled? Explain.

83. CHALLENGE Prove that for an arithm etic sequence, an = ak + {11 k)d for integers k in the dom ain of the sequence.

REASONING D eterm in e w h eth er each sta tem ent is tru e o r fa ls e fo r fin ite arith m etic series. E xp lain .

84. If you know the sum and d, you can solve for av

85. If you only know the first and last term s, then you can find the sum .

86. If the first three term s o f a sequence are positive, then all o f the term s o f the sequence are positive or the sum of the series is positive.

(8 7 ) CHALLENGE C onsider the arithm etic sequence of odd natural num bers.

a. Find S7 and S9.

b. M ake a con jecture about the pattern that results from the sum s of the corresponding arithm etic series.

C. W rite an algebraic proof verify ing the conjecture that you m ade in p art b .

88. WRITING IN MATH Explain w hy the arithm etic series 25 +20 + 15 + . . . does not have a sum .

6 0 6 Lesson 10-2 Arithm etic Sequences and Series

Spiral Review

8 9 .1 2 , 1 6 ,2 0 , . . . 9 0 . 3 , 1 , - ! , . . .

Find the next four terms of each sequence. (Lesson 10-1)

9 1 .3 1 ,2 4 ,1 7 ,

Find each product or q u o tien t and express it in rectan gu lar form . (Lesson 9-5)

92. 6 5 t t , 5 t t cos - + i sin 6 6

93. 3|cos + i sin + |cos y + i sin y j

F ind the dot product o f u and v. T h en d eterm ine i f u and v are orthogonal. (Lesson 8-3)

94. u = (4, 1), v = (1, 5) 95. u = (8 , - 3 ) , v = (4 ,2 ) 96. u = (4, 6 ), v = ( 9 , - 5 )

Find the d irection ang le o f each vector to the n earest ten th o f a degree. (Lesson 8-2)

97. - i - 3j 98. ( - 9 , 5 ) 99. ( - 7 , 7)

100. MANUFACTURING A cam in a punch press is shaped like an ellipse w ithx1 V 2the equation + = 1. The cam shaft goes through the focus on the

positive axis. (Lesson 7-4)

a. G raph a m odel of the cam .

b. Find an equation that translates the m odel so that the cam shaft is at the origin.

C. Find the equation of the m odel in part b w hen the cam is rotated to an upright position.

101. U se the graph of f(x ) = In x to describe the transform ation that results in the graph of g(x) = 3 In (x 1). Then sketch the graphs o f/ and g. (Lesson 3-2)

Skills Review for Standardized Tests

102. SAT/ACT W hat is the units d igit o f 3 36?

A 0

B 1

C 3

D 7 E 9

103. U sing the table, w hich form ula can be used to determ ine the nth term o f the sequence?

F a n = 6n

G u n y i -(- 5

H a n = 2n + 1

J a n = 471 + 2

n

1 6

2 10

3 14

4 18

104. REVIEW If flj = 3, flj = 5, and = an _ 2 + 3n, find a w.

A 59 C 89

B 75 D 125

105. REVIEW W hich of the sequences show n below is convergent?

G - ( 7

H

y

O x

connectED.mc9 raw-hili.com | 6 0 7

Camshaft

You found terms and means of arithmetic sequences and sums of arithmetic series.(Lesson 10-2)

I Find nth terms and geometric means of geometric sequences.

2 Find sums of n terms of geometric series and the sums of infinite geometric series.

The first summer X Games took place in Rhode Island in 1995 and included 27 events. Due to their growing popularity, the winter X Games were introduced at Big Bear Lake, California, in 1997. With an immense fan base, the annual X Games now receive live 24-hour network coverage. Since its inaugural year, the event has seen an average growth of 13% in revenue each year.

NewVocabularygeometric sequence common ratio geometric means geometric series

6 0 8 Lesson 10-3

f Geometric Sequences a sequence in w hich the ratio betw een successive term s is a constant is called a geom etric seq u en ce. The constant is referred to as the com m on ratio,

denoted r. To find the com m on ratio of a geom etric sequence, d ivide any term follow ing the first term by the preceding term . G iven a term of the sequence, to find the next term o f the sequence, m ultiply the given term by the com m on ratio. W hile the rate of change of an arithm etic sequence is constant, the rate of change of a geom etric sequence can either increase or decrease.

JQ 2 E C 3 3 E Geometric SequencesD eterm in e the com m on ratio , and fin d the n ex t three term s o f each geom etric sequ ence,

a. 8, 2, j , . . .

First, find the com m on ratio.

a0 + , = 2 -j- 8 or l l 4

a3 + a 2 = \ ~=~ ~ 2 or j

Find the ratio between two pairs of consecutive terms to verify the common ratio.

1 1 The com m on ratio is M ultip ly the third term by to find the fourth term , and so on.

or3 2 ( 4 ) r 1 2 8

1 1 1The next three term s are , ando 32. 12o

b. w + 3, 2w + 6, 4i + 3 )

w + 3

; 2 w + 6 anda, = w + 3

Factor.

Simplify,

Cl q T" Cl o 4 iv + 1 2 ; 4 w + 12 and

2 2 w + 6

4 (w + 3 ) _ 2 (w + 3 ) = 2

a2 2 w -f~ 6

Factor,

Simplify,

The com m on ratio is 2. M ultip ly the third term b y 2 to find the fourth term , and so on.

ai = 2{4w + 12) or 8w + 24

n5 = 2(8w + 24) or 16w + 48

a6 = 2 (16w + 48) or 32w + 96

The next three term s are 8w + 2 4 ,1 6 w + 48, and 32w + 96.

p GuidedPractice1A. 4 ,1 1 ,3 0 .2 5 , . . . 1B. 64r 128, 16r + 32, 4r 8, ...

WatchOut!Type of Sequence Remember that if a sequence is not arithmetic, it does not necessarily mean that the sequence is geometric. Test several terms for a common ratio before determining that the sequence is indeed geometric.

In Lesson 10-2, you learned that arithm etic sequences can be defined b oth recursively and explicitly. This also applies to geom etric sequences. A geom etric sequence can be expressed recursively, w here a term a is found by taking the product of the previous term an _ x and r, or a n = a n _ j r, as illustrated by the previous exam ple. To develop an exp licit form ula for a geom etric sequence, consider the pattern created b y the geom etric sequence for w hich a l = 3 and r = 4.

Term E xpanded Form E xp o n en tia l Form Exam ple

first term i a \ Bl 3

second term flj r f l jr 1 3 4 = 12

third term a3 flj r r fl jr2 3 4 2 = 48

fourth term 4 flj r r r f l j r 3 3 4 3 = 192

fifth term a5 flj r r r r fljr4

OONOIVIICO

nth term a n flj r r r . . . r _ 1 3 4 " - 1

n 1 factors

KeyConcept The n V n Term of a Geometric SequenceWords The nth term of a geometric sequence with firs t term a-, and common ratio r is given by a n = a^ rn ~ 1.

Example The 9th term of 2 , 10 , 5 0 , . . . is a 9 = 2 5 9 ~ 1 or 781 ,250.L ........ ............................

H ^ S E llEISE Explicit and Recursive FormulasW rite an exp licit fo rm u la and a recursive form u la fo r f in d in g th e nth term o f the geom etric seq u en ce g iv en in Exam ple la .

For an exp licit form ula, substitu te a 1 = 8 and r = 0.25 in the nth term form ula.

a n = f l j f " ~ 1 nth term of a geometric sequence

= 8 (0 .2 5 ) n " 1 a, = 8 and r = -0 .2 5

For a recursive form ula, state the first term a v Then indicate that the next term is the product of the previous term a n _ 1 and r.

flj = 8, a n = (0.25)an _ j

p GuidedPractice2. W rite an exp licit form ula and a recursive form ula for finding the nth term in the sequence

2 ,2 5 ,3 1 2 .5 , . . . .

V ...............................................................................................................................

F inding the nth term of a geom etric sequence is sim plified by explicit form ulas.

F in d the 27th term o f the geom etric seq u en ce 1 8 9 ,1 5 1 .2 ,1 2 0 .9 6 , . . . .

First, find the com m on ratio.

a 2 -j- fli = 151.2 189 or 0.8 Find the ratio between two pairs of consecutive terms to . 1 o n o c . i n n n o verify the common ratio.a3 -r- a 2 = 120.96 151.2 or 0.8

U se the form ula for the nth term of a geom etric sequence.

an - f l jr " 1 n th term of a geometric sequence

a27 = 189(0 .8)27 1 n = 27, a1 = 189, and r = 0.8

a27 ~ 0.57 Simplify.

6 0 9

Fin d the sp ecified term o f each geom etric seq u en ce or seq u en ce w ith the g iven characteristics.

3A. a9 for 4 , 1 4 ,4 9 , . . . 3B. a 12 if a3 = 32 and r = 4

$ GuidedPractice

Just as arithm etic sequences are linear functions w ith restricted dom ains, geom etric sequences are also functions. C onsider the exponential function/(x) = 2 (2)x and the explicit form ula for the geom etric sequence a n = 2 (2 )".

N otice that the graphs of the term s of the geom etric sequence lie on a curve, as show n. A geom etric sequence can be m odeled by an exponential function in w hich the dom ain is restricted to the natural num bers.

a 4\a

C\J

C\JII

20

2 4 n

Real-World Example 4 nth Term of a Geometric Sequence

Real-WorldLinkThe value of a newly purchased vehicle can depreciate by as much as 30-35% in its firs t year. Each year after, the value continues to depreciate by 7-12% , depending on the make and model. After a five-year period, on average, cars are worth 35% of the original sticker price, making car buying costly.

Source: Kelly Blue Book

AUTOMOBILE D am ian pu rchased a la te-m o d el car fo r $15,000. A t the end o f each year, th e valu e o f the car d ep reciates 11%.

a. W rite an exp licit form u la fo r the v alu e o f D am ian 's car a fter n years.

If the ca r 's value depreciates at a rate o f 11% per year, it retains 100% 11% or 89% of its original value. N ote that the original value given represents the a 0 and not the a 1 term , so w e need to use an ad justed form ula for the nth term of this geom etric sequence.

first term n1 = a 0r

second term

nth term

: a nf~

= ClnV

U se this ad justed form ula to find an explicit form ula for the valu e of the car after n years.

a = a nr "

a n = 1 5 ,0 0 0 (0 .8 9 )"

Adjusted /rth term of a geometric sequence

a0 = 15,000, r = 0.89

b. W hat is the valu e o f D am ian 's car at the end o f th e sev en th year?

Evaluate the form ula found in part a for n = 7.

a = 15,000(0.89)" Original equation

= 15,000(0.89)7 n = 7

6634.70 Simplify.

The value of the car at the beginn ing of the seventh year is about $6634.70.

The value of the car at each year is show n as a point on the graph. The function connecting the points represents exponential decay.

GuidedPractice

4. WATERCRAFT Rohan purchased a personal w atercraft for $9000. A ssum e that by the end of each year, the value o f the w atercraft depreciates 30% .

A. W rite an explicit form ula for finding the value o f R ohan 's w atercraft after n years.

B. W hat is the value of R ohan 's w atercraft after 5 years?

6 1 0 | Lesson 10-3 Geometric Sequences and Series

Sim ilar to arithm etic sequences, if tw o nonconsecu tive term s o f a geom etric sequence are know n, the term s betw een them can b e calculated. These term s are called g eo m etric m eans.

StudyTipGeometric Means Sometimes, more than one set of geometric means are possible. For example, the three geometric means between 3 and 48 can be 6,12, and 24 o r - 6,12, and -2 4 .

W rite a seq u en ce that has tw o geom etric m ean s b e tw een 480 and 7.5.

The sequence w ill resem ble 480, ? , __L_, 7.5.

N ote that a x = 480, n = 4, and a4 = 7.5. Find the com m on ratio using the nth term for a geom etric sequence form ula.

- n V * - ia , = a , r nth term of a geometric sequence

- 7 .5 = 480r4 1

"6 4

- H

Simplify and divide each side by 480.

Take the cube root of each side.

1The com m on ratio is . U se r to find the geom etric m eans.

a2 = 480(0.25) or - 1 2 0 a3 = - 1 2 0 ( - 0 .2 5 ) or 30

Therefore, a sequence w ith tw o geom etric m eans betw een 480 and 7.5, is 480, 1 2 0 ,3 0 , 7.5.

GuidedPracticeFin d the ind icated geom etric m ean s fo r each p a ir o f n o n co n secu tiv e term s.

5A. 4 and 13.5; 2 m eans 5B. 10 and 0.016; 3 m eans

i Geometric Series A geom etric series is the sum of the term s of a geom etric sequence.G eo m etric S eq u en ce G eo m etric Series

2 ,4 , 8 ,1 6 ,3 2 2 + 4 + 8 + 16 + 32

2 7 ,9 , 3,

Uy #2/ 3/ ^4/ - /

27 - ! + 3 + l + | -

A form ula for the sum S of the first n term s of a fin ite geom etric series can be developed by looking at the series S and rSn. To create the term s for rSn, each term in Sn is m u ltiplied by r. These series are then aligned so that sim ilar term s are grouped together and then rSn is subtracted from S.

( - ) r S =

S =

tt2 h Cl-yr + f l j r 2 + + f l j r " 2 + a^rn 1

1 r + a xr2 + + f l j r " ~ 2 + axrn ~ 1

f l j f l j r " Subtract.

f l j f l j r Factor,

a 1 a rDivide each side by 1 - r.

1 - r

a1 (1 - r") 1 - r

Factor.

Therefore

^j^connectE^ricgra^^^^^^ 6 1 1

If the value o f n is not p rovided, the sum o f a finite geom etric series can still be found. I f w e take a look at the next-to-last step of the proof, w e can substitu te for a 1r".

S = - 1 - r

1 - r

1 - r

Sum of a fin ite geometric series formula

Multiply.

Factor one /from a / " .

a^rn ~ 1 = a, nth term of a geometric sequence formula

KeyConcept Sum of a Finite Geometric SeriesThe sum of a finite geometric series with n terms or the nth partial sum of a geometric series can be found using one of two related formulas.

Formula 1

Formula 2 1 - r

StudyTipInfinite vs. Finite Geometric Series Notice that the series given in Example 6a is an infinite geometric series. Because you are asked to find the sum of the first six terms of the series, you are actually finding the sum of a finite series.

la r r t f i Sums of Geometric Series

>

a. Find the sum of the first six terms of the geometric series 8 + 14 + 24.5 + . . . .

First, find the com m on ratio.

a2 -r flj = 14 8 or 1.75 Find the ratio between two pairs of consecutive terms toverify the common ratio.

The com m on ratio is 1.75. U se Form ula 1 to find the sum of the series.

a3 a 2 = 24.5 -f 14 or 1.75

Formula 1

and r = 1.75

S 6 a 295.71 Simplify.

The sum o f the first six term s of the geom etric series is about 295.71.

CHECK The next three term s of the related sequence are 42.875, 75.03125, and 131.3046875.

8 + 14 + 24.5 + 42.875 + 75.03125 + 131.3046875 295.71

b. Find the sum o f the first n term s o f a geom etric series w ith a t = 3, = 768, and r = 2.

U se Form ula 2 for the sum o f a finite geom etric series.

1 ~ rr1 - r

3 - 768(2) 1 - ( - 2 )

= 513

Formula 2

a, = 3, a = 768, and r = 2

Simplify.

The sum of the first n term s o f the geom etric series is 513.

^ GuidedPractice6A. Find the sum of the first 11 term s of the geom etric series 7 + (24.5) + 85.75 + . . . .

6B. Find the sum of the first n term s of a geom etric series w ith = 8, a n = 131,072, and r - 4.

6 1 2 Lesson 10-3 I Geometric Sequences and Series

G eom etric series m ay also be represented in sigm a notation.

StudyTipInfinite Series If the sequence of partial sums S has a limit, then the corresponding infinite series has a sum and the nth term a of the series approaches 0 as n oo. However, if the /7th term of the series approaches 0, the infinite series does not necessarily have a sum. For example, the harmonic series 1 + ^ ^ +1 2 3- j + ... does not have 4a sum.

v __________

Geometric Sum in Sigma Notation

Fin d X 3 (5 )" ~ *.n=2

Find n, a l: and r.

n = 7 2 + 1 or 6

= 3(5 ) 2 1 or 15

r = 5

Upper bound minus lower bound plus 1

n = 2

r is the base of the exponential function.

Method 1 Substitu te n = 6 , a : = 15, and r = 5 into Form ula 1.

Method 2

S" = a i (t t )

S 6 = 58, 590

Find a n.

a n = a^r" ~ 1

= 15(5)6- 1

= 46,875

Formula 1

= 15, r = 5, and n = 6

Simplify.

/7th term o f a geometric sequence

a, = 15 , r = 5, and /7 = 6

Simplify.

Substitute flj = 15, a n = 46,875, and r = 5 into Form ula 2.

Formula 2S 1 !n 1 - r

S 6 =15 - (46,875)(5)

1 - 5

S 6 = 58,590

7

Therefore, X 3 (5 ) - 1 = 58,590.n = 2

a , = 1 5, a = 46,875, and r = 5

Simplify.

GuidedPractice31

7A. X 0 -5 1 - 1n = 16

11

7B. X 1 20^ 5 ) - 1n = 4

/ In Lesson 10-1, you learned that calculating the sum s of infinite series m ay be possible if the sequence o f term s converges to 0. For this reason, the sum s of infinite arithm etic series cannot be found.

The form ula for the sum of a finite geom etric series can be used to develop a form ula for the sum of an infinite geom etric series. If \r\ > 1, then | r n \ increases w ithout lim it as n > o o . H ow ever, w hen |r| < 1 , r " approaches 0 as n > o o . Thus,

S =

1 - r

Sum of a fin ite geometric series formula

r" approaches 0 as n * oo.

Simplify and multiply.

KeyConcept The Sum of an Infinite Geometric SeriesThe sum S o f an infinite geometric series for which |r| < 1 is given by

\ .....

connectED.mcgraw-hill.com I 6 1 3

The form ula for the sum of an infinite geom etric series involves three variables: S, a v and r. If any tw o o f the three variables are know n, you can solve for the third.

- 1 < r< 1.

pmETnffBT) Sums of Infinite Geometric Series

StudyTipCommon Ratio Recall that | r | < 1 is equivalent to

I f p o ssib le , fin d the sum o f each in fin ite geom etric series,

a. 9 + 3 + 1 + . . .

First, find the com m on ratio.

a2 + a i = 3 + 9 or i Find the ratio between two pairs o f consecutive terms to verify the common ratio.

3 + a2 = 1 + 3 or

The com m on ratio r is and | i | < 1. This infinite geom etric series has a sum . U se the

form ula for the sum o f an infinite geom etric series.

S =a

1 - r

9

1

Sum of an infin ite geometric series formula

a1 = 9 and r = i

= 13.5

The sum o f the infinite series is 13.5.

b. 0.25 + ( -1 .2 5 ) + 6.25 + . . .

First, find the com m on ratio.

do . tt-\ -1.25 + 0.25 or - 5 Find the ratio between two pairs of consecutive terms to verify the common ratio.

a 3 + a 2 = 6.25 + ( 1.25) or 5

The com m on ratio r is 5, and |5| > 1. Therefore, this infinite geom etric series has no sum.

c. 4(0 .2 ) - 1n=4

The com m on ratio r is 0.2, and |0.2| < 1. Therefore, this infinite geom etric series has a sum.Find a v

Exercises jp = Step-by-Step Solutions begin on page R29.

D eterm ine the com m on ratio , and fin d the next three term s o f each geom etric sequ ence. (Example 1)

1. , , 1, . . .4 ' 2 ' '

O 1 3 9 2 ' 8 '3 2 '" '

3. 0 .5 ,0 .7 5 ,1 .1 2 5 , . . . 4. 8 , 2 0 , 5 0 , . . .

5. 2x, lO x, 5 0 x , . . . 6. 64x, 16x, 4 x ,...

7. x 4- 5 , 3x + 1 5 ,9x + 4 5 , . . .

8 . - 9 - y , 27 + 3i/, - 8 1 - 9 y , . . .

9. GEOMETRY C onsider a sequence o f circles w ith diam eters that form a geom etric sequence: d v d2, d 3, d4, d 5.

a. Show that the sequence of circum ferences of the circles is also geom etric. Identify r.

b. Show that the sequence of areas o f the circles is also geom etric. Identify the com m on ratio.

W rite an exp licit form ula and a recu rsive form u la fo r fin d in g the Hth term o f each geom etric seq u en ce. (Example 2)

10. 3 6 ,1 2 ,4 , . . .

12. - 2 , 1 0 , - 5 0 , . . .

14. 4 , 8 ,1 6 , . . .

16. 1 5 ,5 , | , . . .

11. 6 4 ,1 6 ,4 , . . .

13. 4 , - 1 2 , 3 6 , . . .

15. 2 0 ,3 0 ,4 5 , . . .

17. . . .32' 16' 8 '

18. CHAIN E-MAIL M elina receives a chain e-m ail that she forw ards to 7 of her friends. E ach of h er friends forw ards it to 7 o f their friends. (Example 2)

C hain E-Mail

a. W rite an explicit form ula for the pattern.

b. H ow m any w ill receive the e-m ail after 6 forw ards?

(1 9 ) BIOLOGY A certain bacteria d ivides every 15 m inutes to produce tw o com plete bacteria. (Example 2)

a. If an initial colony contains a population o f b0 bacteria, w rite an equation that w ill determ ine the num ber of bacteria b t present after t hours.

b. Suppose a Petri d ish contains 12 bacteria. U se the equation found in part a to determ ine the num ber of bacteria present 4 hours later.

F ind the sp ecified term fo r each geom etric seq u ence or seq u en ce w ith the g iv en ch aracteristics . (Example 3)

20. a 9 for 6 0 ,3 0 ,1 5 , . . .

22. a . for 3 ,1 , . . .

21. a4 for 7 ,1 4 ,2 8 , . . .

23. a6 for 5 4 0 ,9 0 ,1 5 , . . .

24. #7 if a 3 = 24 and r = 0.5 25. ab if a3 = 32 and r = 0.5

26. a6 if flj = 16,807 and r = ~ 27. a8 if a x = 4096 and r = j

28. ACCOUNTING Ju lian Rockm an is an accountant for a sm all com pany. O n Janu ary 1, 2009, the com pany purchased $50,000 w orth o f com puters, printers, scanners, and hardw are. Because this equipm ent is a com pany asset, Mr. Rockm an needs to determ ine how m uch the com puter equipm ent is p resently w orth. H e estim ates that the com p uter equipm ent depreciates at a rate of 45% per year. W hat value should Mr. Rockm an assign the equipm ent in his 2014 year-end accounting report? 'Example 4)

29. F ind the sixth term of a geom etric sequence w ith a first term o f 9 and a com m on ratio o f 2. (Example 4)

30. If r = 4 and a8 = 100, w hat is the first term of the geom etric sequence? (Example 4)

31. X GAMES Refer to the beginn ing o f the lesson. TheX G am es netted approxim ately $40 m illion in revenue in 2002. If the X G am es continue to generate 13% m ore revenue each year, how m u ch revenue w ill the X Gam es generate in 2020? (Example 4)

Find the in d icated geom etric m ean s fo r each p a ir o f no n co n secu tiv e term s. (Example 5)

32. 4 and 256; 2 m eans

34. y and 7; 1 m ean

36. 1 and 27; 2 m eans

38. i and 1; 4 m eans

33. 256 and 81; 3 m eans

35. 2 and 54; 2 m eans

37. 48 and 750; 2 m eans

39. f8 and t~7; 4 m eans

F ind the sum o f each geom etric series d escribed . (Example 6)

40. first six term s of 3 + 9 + 27 +

41. first n ine term s o f 0.5 + (1) + 2 +

42. first eight term s of 2 + 2\/3 + 6 +

43. first n term s o f a x = 4, a n = 2000, r = 3

= 5, a n = 1,747,625, r = 4

= 3, a n = 46,875, r = 5

= 8, a n = 256, r = 2

= - 3 6 , a = 972, r = 7

44. first n term s o f a

45. first n term s o f a

46. first n term s o f a

47. first n term s o f a

6 1 5

Find each sum. (Example 7)6

4 8 . E 5 " - 1n = 1

5

4 9 . E - 4 ( 3 ) nn = 1

5 0 . E ( - 3 ) " _ 1n = 1

6

5 1 . E 2 d - 4 ) "n = 1

5 2 . 1 0 0 ( f ) - 1n = 1

5 3 . E - L ( - 3 )n = l z /

5 4 . E l 4 4 ( - I ) " _ 1n = 1 v z /

2 0

5 5 . E 3 ( 2 ) " - :n = 1

I f p o ssib le , find the sum o f each in fin ite geom etric series. (Example 8)

EC 1 I 1 . 1 , C7 2 . 4 . 8 .2 0 4 0 8 0 7 7 7

58. 18 + ( - 2 7 ) + 40.5 + 59. 12 + ( - 7 .2 ) + 4.32 + OO OO / \ 1

60. E 6 (0.4)" - 1 61 . E 40(| )

64. BUNGEE JUMPING A bungee jum per falls 35 m eters before2

his cord causes him to spring back up. H e rebounds of the distance after each fall. (Example 8)

a. Find the first five term s of the infinite sequence representing the vertical distance traveled by the bungee jum per. Include each drop and rebound distance as separate terms.

b. W hat is the total vertical distance the ju m p er travels before com ing to rest? (Hint: Rew rite the infinite sequence suggested b y part a as tw o infinite geom etric sequences.)

Find the m issin g qu an tity fo r the geom etric seq u en ce w ith the g iven characteristics.

65. Find a 1 if S 12 = 1365 and r = 2.

66. If S 6 = 196.875, a l = 100, r = 0.5, find a b.

67. Find r if = 0.12, Sn = 590.52, and a = 787.32.

68. Find n for 4.1 + 8 .2 + 16.4 + - if Sn = 61.5.

69. If 1 5 - 1 8 + 2 1 .6 ------- , S = 23.784, find a n.

70. If r = 0.4, S 5 = 144.32, and a x = 200, find a5.

71. Find a 1 if S = 468, a = 375, and r = 5.

72. I f S = , | + i + | + - , f i n d n .

73. LOANS M arc is m aking m onthly paym ents on a loan. Suppose instead of the sam e m onthly paym ent, the bank requires a low initial paym ent that grow s at an exponential rate each m onth. The total cost o f the loan is

k

represented by E 5 (1 .1 )" - 1 .=i

a. W hat is M arc's initial paym ent and at w hat rate is this paym ent increasing?

b. If the sum of M arc's paym ents at the end of the loan is $7052, how m any paym ents did M arc m ake?

6 1 6 Lesson 10-3 Geom etric Sequences and Series

Find the com m on ratio fo r the geom etric seq u en ce w ith the g iv en term s.

74. a3 = 12, a6 = 187.5

75. a2 = 6, a7 = 192

76. a4 = - 2 8 , a6 = - 1 3 7 2

77. a5 = 6 , fl8 = 0.048

78. ADVERTISING W ord-of-m outh advertising can be an effective form of m arketing, or it can be very harm ful. C onsider a new restaurant that serves 27 custom ers on its opening night.

a. O f the 27 custom ers, 25 found the experience enjoyable and each told 3 friends over the next m onth. This group each told 3 friends over the next m onth, and so on, for 6 m onths. A ssum ing that no one heard tw ice, how m any people have had a positive experience or heard p ositive review s of the restaurant?

b. Supp ose the 2 unhappy custom ers each told 6 friends over the next m onth about the experience. This group then each told 6 friends, and so on, for 6 m onths. A ssum ing that no one heard a review tw ice, how m any people have had a negative experience or heard a negative review ?

W rite the first 3 term s o f th e in fin ite geom etric series w ith the g iv en characteristics.

79. S = 12, r = f 80. S = - 2 5 , r = 0.2

81. S = 4 4 .8 ,fl1 = 56 82. S = | , f l ! = |

83. S = - 6 0 , r = 0.4 84. S = -1 2 6 .2 5 , a x = - 5 0 .5

85. S = -1 1 5 , a 1 = - 1 3 8 86. S = r =

oo . . OO , . -t88- l ( t )

D eterm in e w h eth er each seq u en ce is arithm etic, geom etric, or neither. T h en fin d the next three term s o f the sequ ence.

8p 1 2 3 4 qn 9 17 . 158 9 ' 4 ' 6 ' 8 ' 1 0 ' - 9 - 2 ' T ' 4 ' T ' -

91 . 1 2 ,2 4 , 3 6 ,4 8 , . . . 92. 1 2 8 ,9 6 , 72, 5 4 , . . .

93. 36k, 49k, 64k, 81 k , . . . 94. 7.2y, 9.1y, l ly , 1 2 .9 y , . . .

95. 3 V 5 , 1 5 ,1 5 V 5 , 7 5 , . . . 96. 2 \ / 3 , 2 ^ 6 , 2 V 9 , 2\fV2.,...

W rite each geom etric series in s igm a notation .

9 7 .3 + 12 + 48 + + 3072

98. 9 + 18 + 36 + - + 1152

99. 50 + 85 + 144.5 + + 417.605

100. 1 - I + I + 88 4 2

101. 0.2 - 1 + 5 ----------- 625

102. HORSES For each of the first few m onths after a horse is born, the am ount o f w eight that it gains is about 120% of the previous m on th 's w eight gain. In the first m onth, a horse has gained 30 pounds.

a. W rite a geom etric series in sigm a notation that can be used to m odel the horse 's w eight gain for the first five m onths.

b. A bout how m uch w eight d id the horse gain in the fourth m onth?

c. If the horse w eighed 150 pounds at b irth , about how m uch did it w eigh after 5 m onths?

d. W ill the horse continue to grow at this rate indefinitely? Explain.

103. MEDICINE A new ly developed and researched m edicine has a half-life of about 1.5 hours after it is adm inistered. The m edicine is given to patients in doses of d m illigram s every 6 hours.

a. W hat fraction of the first dose w ill be left in the p atient's system w hen the second dose is taken?

b. Find the first four term s o f the sequence that represents the am ount of m edicine in the p atient's system after the first 4 doses.

c. W rite a recursive form ula that can be used to determ ine the am ount of m edicine in the patien t's system after the nth dose.

104. 5 MULTIPLE REPRESENTATIONS In this problem , you w ill1 rninvestigate the lim its of .

a. GRAPHICAL G raph S = 1 ~ for r = 0.2, 0.5, and 0.9 on the sam e graph.

b. TABULAR C opy and com plete the table show n below .

n S ,r= 0.2 Sn, r 0.5 Sm r 0.90

4

8

12

16

20

24

C. ANALYTICAL For each graph in part a, describe the values of S as n > oo.

d. GRAPHICAL G raph S = 1 ~ '' for r = 1 .2 ,2 .5 , and 4 on the sam e graph.

e. ANALYTICAL For each graph in part d , describe the values of Sn as n oo.

f. ANALYTICAL M ake a conjecture about w hat happens to

C C c 1 - 8 . 6 "S as n oo for Sn = 1 _ 8 6 -

H.O.T. Problems Use Higher-Order Thinking Skills(105) ERROR ANALYSIS E m ilio believes that the sum of the

infinite geom etric series 16 + 4 + 1 + 0.25 + ... can be calculated. A nnie disagrees. Is either of them correct? Explain your reasoning.

106. CHALLENGE A b all is dropped from a height o f 5 m eters. O n each bounce, the ball rises to 65% of the height it reached on the previous bounce.

A X 43.25 m

!

13.25 miiT

a.

b.

A pproxim ate the total vertical d istance the ball travels, until it stops bouncing.

The ball m akes its first com plete bounce in 2 seconds, that is, from the m om ent it first touches the ground until it n ext touches the ground. Each com plete bounce that follow s takes 0.8 tim es as long as the preceding bounce. E stim ate the total am ount of tim e that the ball bounces.

107. WRITING IN MATH Explain w hy an infinite geom etric series w ill not have a sum if \r\ > 1.

REASONING D eterm in e w h eth er each sta tem en t is true or fa l s e . E xp lain you r reasoning .

108. If the first tw o term s of a geom etric sequence are positive, then the third term is positive.

109. If you know r and the sum o f a finite geom etric series, you can find the last term .

110. If r is negative, then the geom etric sequence converges.

111. REASONING D eterm ine w hether the follow ing statem ent is sometimes, always, or never true. Explain your reasoning.If all of the terms of an infinite geometric series are negative, then the series has a sum that is a negative number.

112. CHALLENGE The m idpoints of the sides of a square are connected so that a new square is form ed. Suppose this process is repeated indefinitely.

a. W hat is the perim eter of the square w ith side lengths o f x inches?

b. W hat is the sum of the perim eters of all the squares?c. W hat is the sum of the areas o f all the squares?

$conne^EDj^grav!ni!ll.com| 6 1 7

Spiral Review

Find each sum . ;Lesson 10-2)7 7 1 5 0

113. j2 ( 2 n + l) 114. X ) ( 3 + 4) 115. (11 + 2n)n = 1 w = 3 r c = l

116. TOURIST ATTRACTIONS To prove that objects of d ifferent w eights fall at the sam e rate, M arlene dropped tw o objects w ith different w eights from the Lean ing Tow er of Pisa in Italy. The objects hit the ground at the sam e tim e. W hen an ob ject is dropped from a tall b u ild ing , it falls about 16 feet in the first second, 48 feet in the second second, and 80 feet in the third second, regardless of its w eight. If this p attern continues, how m any feet w ould an object fall in the sixth second? (Lesson 10-1)

117. TEXTILES Patterns in fabric can often be created by m odifying a m athem atical graph.The pattern at the right can be m odeled by a lem niscate. Lesson 9-2)

a. Suppose the designer w anted to begin w ith a lem niscate that w as 6 units from end to end. W hat polar equation could have been used?

b. W hat polar equation could have been used to generate a lem niscate that w as 8 units from end to end?

G raph each p olar eq u atio n on a p o lar grid . (Lesson 9-1)

118. 9 = 119. r = 1.5 120. 9 = - 1 5 0 4

Find the cross product o f u and v. T h en show that u X v is orthog onal to b o th u and v.(Lesson 8-5)

121. u = (1 ,9 , - 1 ) , v = ( - 2 , 6, - 4 } 122. u = ( - 3 , 8 ,2 ) , v = (1, - 5 , - 7 ) 123. u = (9, 0, - 4 ) , v = ( - 6, 2 , 5 )

Find the com ponent form and m agnitud e o f A B w ith the g iv en in itia l and term in al po in ts.T hen fin d a u n it vector in the d irection o f A B . (Lesson 8-4)

124. A(6, 7 ,9 ) , B (1 8 ,2 1 ,1 8 ) 125. A (24, - 6 ,1 6 ) , B(8,1 2 , - 4 ) 126. A(3, - 5 , 9 ) , B ( - 1 ,1 5 , - 7 )

Skills Review for Standardized Tests

127. SAT/ACT In the geom etric sequence , j , . . . , eachterm after the first is equal to the previous term tim es a constant. W hat is the value of the 13 th term ?

A 2 7

B 2 8

C 2 9

D 2 io

E 2 u

128. REVIEW The pattern o f dots show n below continues infinitely, w ith m ore dots being added at each step.

O O O O O O O O OO O O O O O O O O

FirstStep

SecondStep

ThirdStep

W hich expression can be used to determ ine the num ber of dots at the nth step?

F 2n H n(n + 1)

G n(n + 2) J 2 (n + 1)

129. The first term of a geom etric series is 1, and the com m on ratio is 3. H ow m any term s are in the series if its sum is 182?

A 6

B 7

C 8

D 9

130. REVIEW C ora begins a phone tree to notify her friends about a party. In step 1, she calls 3 friends. In step 2, each of those friends calls 3 new friends. In step 3, each of those friends calls 3 m ore new friends. A fter step 3, how m any people know about the party, inclu ding Cora?

F 12

G 13

H 39

J 40

6 1 8 Lesson 10-3 | Geometric Sequences and Series

m r . u i u v i l w w ^ p

Graphing Technology LabContinued Fractions

O O O Oooooooooo o o

Use a graphing calculator to represent continued fractions.

StudyTipMemory You may need to clear the calculators memory to eliminate any previously stored values.

An expression of the following form is called a continued fraction. Continued fractions can be used to write sequences that approach limits.

bu + -

StudyTipFinding Expressions When developing the expression in 2C, think of different ways in which b can make the expression equivalent to the lim it of the sequence. Use different values of