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Geometry Suppose you are stacking boxes in levels that form squares. Th e numbers of boxes in successive levels form a sequence. Th e fi gure at the right shows the top four levels as viewed from above.
a. How many boxes of equal size would you need for the next lower level?
b. How many boxes of equal size would you need to add three levels?
c. Suppose you are stacking a total of 285 boxes. How many levels will you have?
1. How many boxes are in each of the fi rst four levels?
Level 1: z z Level 2: z z Level 3: z z Level 4: z z
2. How many boxes of equal size would you need for the next lower level?
3. What is a recursive or explicit formula that describes the number of boxes in the nth level?
4. How many boxes would you need to add three levels?
z z 1 z z 1 z z 5 z z
5. What is a recursive or explicit formula that describes the total number of boxes in a stack of n levels?
6. How can you use your formula to fi nd the number of levels you will have with a stack of 285 boxes?
.
7. Suppose you are stacking a total of 285 boxes. Use your formula to fi nd how many levels you will have. Show your work.
8. You need z z levels to make a stack of 285 boxes.
Determine whether each formula is explicit or recursive. Th en fi nd the fi rst fi ve terms of each sequence.
38. an 513 n 39. an 5 n2 2 6 40. a1 5 5, an 5 3an21 2 7
41. an 512
(n 2 1) 42. a1 5 5, an 5 3 2 an21 43. a1 5 24, an 5 2an21
44. Error Analysis Your friend says the explicit formula for the sequence 1, 8, 27, 64 is an 5 n2. Is she correct? Explain.
45. Writing Explain how to fi nd an explicit formula for a sequence.
46. Th e fi rst fi gure of a fractal contains one segment. For each successive fi gure, six segments replace each segment.
a. How many segments are in each of the fi rst four fi gures of the sequence? b. Write a recursive defi nition for the sequence.
47. Th e sum of the measures of the exterior angles of any polygon is 3608. All the angles have the same measure in a regular polygon.
a. Find the measure of one exterior angle in a regular hexagon (six angles). b. Write an explicit formula for the measure of one exterior angle in a regular
polygon with n angles. c. Why would this formula not be meaningful for n 5 1 or n 5 2?
48. Reasoning In order to fi nd a term in a sequence, its position in the sequence is doubled and then two is added. What are the fi rst ten terms in the sequence?
49. Writing Explain the diff erence between a recursive and an explicit formula.
50. Open-Ended Write fi ve terms in a sequence. Describe the sequence using a recursive or explicit formula.
9-1 Practice (continued) Form G
Mathematical Patterns
explicit; 13, 23, 1, 43, 53
explicit; 0, 12, 1, 1 12, 2
explicit; 25, 22, 3, 10, 19
recursive; 5, 22, 5, 22, 5
recursive; 5, 8, 17, 44, 125
recursive; 24, 28, 216, 232, 264
She is incorrect; in order to fi nd each term
Look for a pattern in the
in the sequence, the term number must be cubed, not squared.
sequence and fi nd a mathematical rule that gives the nth term, given the number n.
An explicit formula defi nes how to fi nd the nth term directly from the number n, while a recursive formula defi nes how to fi nd each term from the previous term(s).
10. 214, 28, 22, 4, 10, c 11. 6, 5.7, 5.4, 5.1, 4.8, c 12. 1, 22, 4, 28, 16, c
13. 1, 3, 9, 27, c 14. 1, 12, 14, 18, 116, c 15. 2
3, 1, 113, 12
3, 2, c
16. 36, 39, 42, 45, 48, c 17. 36, 30, 24, 18, 12, c 18. 9.6, 4.8, 2.4, 1.2, 0.6, c
Write an explicit formula for each sequence. Find the twentieth term.
19. 7, 14, 21, 28, 35, c 20. 2, 8, 14, 20, 26, c 21. 5, 6, 7, 8, 9, c
22. 21, 0, 1, 2, 3, c 23. 3, 5, 7, 9, 11, c 24. 0.8, 1.6, 2.4, 3.2, 4, c
25. 14, 12, 34, 1, 54, c 26. 1
2, 14, 16, 18, 110, c 27. 2
3, 123, 22
3, 323, 42
3, c
Find the eighth term of each sequence.
28. 1, 3, 5, 7, 9, c 29. 400, 200, 100, 50, 25, c 30. 0, 22, 24, 26, 28, c
31. 1, 2, 4, 8, 16, c 32. 44, 39, 34, 29, 24, c 33. 0.7, 0.8, 0.9, 1.0, 1.1, c
34. 4, 11, 18, 25, 32, c 35. 114, 2
12, 5, 10, 20, c 36. 26, 29, 212, 215, 218,
c
37. A man swims 1.5 mi on Monday, 1.6 mi on Tuesday, 1.8 mi on Wednesday, 2.1 mi on Th ursday, and 2.5 mi on Friday. If the pattern continues, how many miles will he swim on Saturday?
Write an explicit formula for each sequence. Th en fi nd the tenth term.
13. 7, 10, 13, 16, c 14. 8, 9, 10, 11, 12, c 15. 212, 0, 12, 1, 1
12, c
an 5 3n 1 4
a10 5 3(10) 1 4 5 z z
16. 1, 4, 9, 16, c 17. 3, 1, 21, 23, 25, c 18. 1, 7, 25, 79, 241
19. Reasoning You and your friend are trying to fi nd the 80th term in the sequence 8, 14, 20, 26, 32, c. You use a recursive defi nition and your friend uses an explicit formula. Who will fi nd the 80th term fi rst? Why?
20. Your neighbor recently began learning to play the guitar. On the fi rst day, she practiced for 0.4 h. On the second day, she practiced for 0.5 h. She practiced for 0.65 h on the third day, and 0.85 h on the fourth day. If this pattern continues, how long will she practice on the seventh day?
21. Charles lost two rented movies, so he owes the rental store a fee of $40. At the end of each month, the amount that Charles owes will increase by 5%, plus a $2 billing fee. How much money will Charles owe the rental store after 8 months?
9-1 Practice (continued) Form K
Mathematical Patterns
an 5 n2; 100
1.75 h
$78.20
Your friend will fi nd the 80th term fi rst because he is using an explicit formula. Your friend will substitute 80 into the formula to get the answer, while you will go through 79 iterations of the recursive formula.
Th e fi rst fi ve terms are 3, 7, z z, z z, and z z.
3. a 512
n 1 2 4. an 5 3n 5. an 5 26n2
6. Write an explicit formula for a sequence with 3, 5, 7, 9, and 11 as its fi rst fi ve terms.
Write a recursive defi nition for each sequence.
7. 2, 6, 12, 20, c 8. 120, 60, 30, 15, c
Identify the initial condition.
a1 5 2
Use n to express the relationship between successive terms.
9. 3, 8, 13, 18, c 10. 1, 3, 9, 27, c 11. 2, 3, 8, 63, c
12. Writing Explain the diff erence between a recursive defi nition and an explicit formula.
2.5, 3, 3.5, 4, 4.5
a1 5 3; an11 5 an 1 5
A recursive formula defi nes a sequence by the relationship between successive terms.An explicit formula describes the nth term of a sequence using the number n.
You can defi ne the terms in a sequence using an explicit formula or a recursive defi nition. You can use another method, called iteration, to form a sequence. Th e word iteration means to repeat an action. In mathematics, a sequence of numbers is generated through iteration when the same procedure is performed on each output.
1. Consider the function f (x) 5 5x 1 1. Let the fi rst term of a sequence be 0. What is f (0)? Let f (0) be the second term of the sequence. Write the sequence.
2. To create more terms of this sequence through iteration, continue to apply f (x) to each output. Th e third term in this sequence can be described as f ( f (0)). What is the third term?
3. Determine the fi rst 10 terms of this sequence. You already have the fi rst 3 terms.
4. Determine the fi rst 5 terms of the sequence formed through iterations off (x) 5
x2 1 1. Begin with x 5 2. Describe the sequence.
5. Will you get the same type of sequence if you start with a diff erent number?
6. Iterations have uses other than to form numerical sequences. Consider this iterative process, which forms a sequence of a set of three integers. Make a set of any three integers. Compute the absolute value of the diff erence between each pair of integers in the set. Th is produces a new set of three integers. Continue this process on each new set of three integers. Describe what eventually happens.
7. You can form fractals through iterations. Fractals are geometric fi gures just like circles or rectangles, but fractals have a special property that these geometric fi gures do not. You make fractals by iterating the fi gure itself. For example, start by drawing an equilateral triangle on graph paper. Divide each side into three equal parts. Draw another equilateral triangle on one side of the triangle that has the middle section as its base. Repeat this process on the remaining two sides. You have just created the fi rst two iterations of a fractal called the Koch snowfl ake.
4. Th e sequence 4, 16, 36, 64, 100, ccan best be represented by which formula?
an 5 4n an 5 4n2 an 5 4n3 an 5 2n4
5. For the sequence 0, 6, 16, 30, 48, c , what is the 40th term?
3198 3200 4000 16,000
6. A student sets up a savings plan to transfer money from his checking account to his savings account. Th e fi rst week $10 is transferred, the second week $12 is transferred, the third week $16 is transferred, and the fourth week $24 is transferred. If this pattern continues and he starts with $100 in his checking account, how many weeks will pass before his balance is zero?
4 5 6 7
Short Response 7. After training for and running a marathon, an athlete wants to reduce her daily run
by half each day. Th e marathon is about 26 mi. How many days will it take after the marathon before she runs less than a mile a day? Show your work.
9-1 Standardized Test Prep Mathematical Patterns
C
G
D
G
A
G
[2] 5 days; Day 1: 13 mi, Day 2: 6.5 mi, Day 3: 3.25 mi, Day 4: 1.625 mi, Day 5: 0.8125 mi [1] correct answer, without work shown OR incorrect answer with correct sequence[0] incorrect answers and no work shown OR no answers given
1. Each term is increased by one more than the previous term; 20, 26, 332. Each term is multiplied by 2 to get the next term; 96, 192, 3843. Each term is multiplied by 22 to get the next term; 64, 2128, 256
4. Each term is multiplied by 3 to get the next term; 243, 729, 21875. Each term is decreased by 5 to get the next term; 75, 70, 656. Each term is increased by 3 to get to the next term; 30, 33, 36
7. Each term is multiplied by 5 to get the next term; 15,625; 78,125; 390,6258. Each term is decreased by one more than the previous term; 35, 29, 229. Each term is divided by 2 to get the next term; 7.5, 3.75, 1.875
10. In each term, the number is increased by two more than the previous term; 33, 45, 5911. Each term is multiplied by 21.5 to get the next term; 607.5, 2911.25, 1366.87512. Each term is multiplied by 2 and then 3 is added to get the next term; 125, 253, 509
Transportation Suppose a trolley stops at a certain intersection every 14 min. Th e fi rst trolley of the day gets to the stop at 6:43 a.m. How long do you have to wait for a trolley if you get to the stop at 8:15 a.m.? At 3:20 p.m.?
Know
1. If you defi ne 12:00 a.m. as minute 0, then 6:43 a.m. is z z from 0.
2. 8:15 a.m. is z z from 0 and 3:20 p.m. is z z from 0.
3. Th e trolley stops every z z.
Need
4. To solve the problem I need to fi nd:
.
Plan
5. What is an explicit formula for the number of minutes after 12:00 a.m. that the trolley gets to the stop?
6. Use your formula to fi nd the smallest n that gives the minutes just after 8:15 a.m. that the trolley arrives at the stop.
7. Using this n in your formula, when does the trolley stop? How long do you have to wait for this trolley?
8. Use your formula to fi nd the smallest n that gives the minutes just after 3:20 p.m. that the trolley arrives at the stop.
9. Using this n in your formula, when does the trolley stop?How long do you have to wait for this trolley?
an 5 403 1 (n 2 1)14
403 min
495 min 920 min
14 min
8
at 501 min
6 min
38
at 921 min1 min
the closest times that the trolley gets to the stop that are after 8:15 A.M.
Determine whether each sequence is arithmetic. If so, identify the common diff erence.
1. 1, 4, 7, 10, c 2. 6, 10, 14, 18, 22, c 3. 1, 3, 6, 10, 15, c
4 2 1 5 3
7 2 4 5 3
10 2 7 5 3
Th is sequence is arithmetic.
Th e common diff erence is z z.
4. 216, 213, 29, 24, 2, c 5. 2, 9, 16, 23, 30, c 6. 43, 56, 69, 82, c
7. Reasoning Is the sequence represented by the formula an 5 4n 1 8 arithmetic? Explain.
Find the 24th term of each arithmetic sequence.
8. 4, 6, 8, 10, 12, c 9. 2, 5, 8, 11, 14, c 10. 9, 5, 1, 23, 27, c
an 5 a1 1 (n 2 1)d an 5 a1 1 (n 2 1)d
a24 5 4 1 (24 2 1)2
a24 5 4 1 46
a24 5 z z
Find the missing terms in the following arithmetic sequences.
11. 2, ___, ___, 14, c 12. 3, z z, z z, 21, c 13. 65, z z, z z, 32, c
14 5 2 1 3d
12 5 3d
d 5 4
2 1 4 5 z z6 1 4 5 z z
14. Error Analysis Noah used the formula an 5 a 1 (n 2 1)d to fi nd the 12th term in the sequence 2, 4, 7, 11, 16, c. Did Noah fi nd the correct term? How do you know?
not arithmetic
Yes; the difference between consecutive terms is 4.
3
50
6
10
9 15 54 43
arithmetic; 7
arithmetic; 4
71
sequence that is not arithmetic.No; Noah applied the explicit formula for arithmetic sequences to a
39. an21 5 y 2 z, an11 5 y 40. an21 5 2t 1 3, an11 5 4t 2 1
41. Open-Ended Write an arithmetic sequence of at least fi ve terms with a positive common diff erence.
42. Error Analysis On your homework, you write that the missing term in the arithmetic sequence 31, ___, 41, c is 351
2. Your friend says the missing term is 36. Who is correct? What mistake was made?
43. Reasoning Explain why 84 is the missing term in the sequence 89, 86.5, ___, 81.5, c.
44. Writing Describe the general process of fi nding a missing term in an arithmetic sequence.
45. You are making an arrangement of cubes in concentric rings for a sculpture. Th e number of cubes in each ring follows the pattern below.
1, 9, 17, 25, 33, c
a. Is this an arithmetic sequence? Explain. b. What are the next three terms? c. If the sequence continues to the 100th term in this pattern, what will that term be?
46. Each year, a volunteer organization expects to add 5 more people to the number of shut-ins for whom the group provides home maintenance services. Th is year, the organization provides the service for 32 people.
a. Write a recursive formula for the number of people the organization expects to serve each year.
b. Write the fi rst fi ve terms of the sequence. c. Write an explicit formula for the number of people the organization expects
to serve each year. d. How many people would the organization expect to serve in the 20th year?
9-2 Practice (continued) Form G
Arithmetic Sequences
108
28.5 1.6
y 2 z2 3t 1 1
a fi ve-term sequence with a positive commondifference
Your friend is correct. You did not takethe average of 31 and 41 correctly to fi nd the missing term of 36.
The common difference in the arithmetic sequence is 22.5, which means the missing term must be 84 as that is 2.5 less than the term before it and 2.5 more than the term after it.
If the term that is missing occurs between two other terms that are consecutive to the missing term, you can take the arithmetic mean of the two terms. If the term that is missing is not consecutive, use the formula an 5 a 1 (n 2 1)d.
Determine whether each sequence is arithmetic. If so, identify the common diff erence.
1. 2, 3, 5, 8, c 2. 0, 23, 26, 29, c
3. 0.9, 0.5, 0.1, 20.3, c 4. 3, 8, 13, 18, . . .
5. 14, 215, 244, 273, c 6. 3.2, 3.5, 3.8, 4.1, c
7. 234, 228, 222, 216, c 8. 2.3, 2.5, 2.7, 2.9, c
9. 127, 140, 153, 166, c 10. 11, 13, 17, 25, c
Find the 43rd term of each sequence.
11. 12, 14, 16, 18, c 12. 13.1, 3.1, 26.9, 216.9, c
13. 19.5, 19.9, 20.3, 20.7, c 14. 27, 24, 21, 18, c
15. 2, 13, 24, 35, c 16. 21, 15, 9, 3, . . .
17. 1.3, 1.4, 1.5, 1.6, c 18. 22.1, 22.3, 22.5, 22.7, c
19. 45, 48, 51, 54, c 20. 20.073, 20.081, 20.089, c
Find the missing term of each arithmetic sequence.
21. c 23, 7 , 49, c 22. 14, 7 , 28, c 23. c 29, 7 , 33, c
24. c 14, 7 , 15, c 25. c 245, 7 , 239, c 26. c 25, 7 , 22, c
27. 22, 7 , 2, c 28. c 26, 7 , 2, c 29. 234, 7 , 77, c
30. c 245, 7 , 212, c 31. 22, 7 , 456, c 32. c 34, 7 , 345, c
33. A teacher donates the same amount of money each year to help protect the rainforest. At the end of the second year, she has donated enough money to protect 8 acres. At the end of the third year, she has donated enough money to protect 12 acres. How many acres will the teacher’s donations protect at the end of the tenth year?
34. Writing Explain how you know that the sequence 109, 105, 101, 97, 93, cis arithmetic.
no yes; 23
yes; 20.4 yes; 5
yes; 229 yes; 0.3
yes; 6 yes; 0.2
yes; 13
96
36.3
464
5.5
171
no
2406.9
299
2231
210.5
20.409
The sequence has a common difference between terms of 24.
Find the missing term of each arithmetic sequence.
15. c4, ___, 18, c 16. c9, z z , 37, c
Find the arithmetic mean of the given terms.
4 1 18 5 22
22 4 2 5 11
Th e missing term is z z .
17. 46, z z , 28, c 18. 212, z z , 24, c 19. c4, z z , 244, c
20. Error Analysis Your friend used the arithmetic mean to fi nd the missing term in the following sequence: 3, ___, 29, 42, c. His answer was 13. What error did your friend make? What is the correct answer?
21. An architect is designing a building with sides in the shape of a trapezoid. Th e number of windows on each fl oor forms an arithmetic sequence. Th ere are 124 windows on the fi rst fl oor and 116 windows on the second fl oor.
a. Write an explicit formula to represent the sequence. b. How many windows are on the tenth fl oor?
22. Your cousin opened a bank account with a deposit of $256 dollars. After one week, she had $280 in her account. After two weeks, she had $304, and after three weeks she had $328. If this pattern continues, how much money will your cousin have in her account after 18 weeks?
23. Th ere is a puddle 1.4 cm deep in your backyard. After one minute of rain, the puddle was 1.45 cm deep. Th e puddle was 1.5 cm deep after it rained for two minutes. If the pattern continues, how deep will the puddle be after it rains for 45 min?
9-2 Practice (continued) Form K
Arithmetic Sequences
11
23
37
He subtracted 3 from 29 when he should have added 3 and 29; 16
2. An arithmetic sequence begins 4, 9, c. What is the 20th term?
76 80 84 99
3. What are the missing terms of the arithmetic sequence 5, __, __, 62, c?
19, 24 19, 34 24, 43 43, 62
4. What is the missing term of the arithmetic sequence 25, __, 45, c?
30 35 37 40
5. Th e seventh and ninth terms of an arithmetic sequence are 197 and 173. What is the eighth term?
161 180 185 221
6. An artist is creating a tile mosaic. She uses 4 green tiles in the fi rst row, 11 green tiles in the second row, 18 green tiles in the third row, and 25 green tiles in the fourth row. If she continues the pattern, how many green tiles will she use in the 20th row?
32 58 134 137
Extended Response
7. What is the 100th term in the arithmetic sequence beginning with 3, 19, c? Show your work.
9-2 Standardized Test Prep Arithmetic Sequences
[4] 1587; a 5 3, n 5 100, d 5 16, an 5 a 1 (n 2 1)d; a100 5 3 1 (100 2 1)16 5 3 1 1584 5 1587
[3] appropriate method shown, with one computational error[2] appropriate method shown, with several computational errors OR correct term found
incorrectly with work shown[1] incorrect term, without work shown[0] incorrect answers and no work shown OR no answers given
Th ere are many types of sequences. One interesting type of sequence is the Farey sequence. Th e fi rst four Farey sequences are:
F1: e 01 , 1
1 f F2: e 0
1 , 12 , 1
1 f F3: e 0
1 , 13 , 1
2 , 23 ,1
1 f F4: e 0
1 , 14 , 1
3 , 12 , 2
3 , 34 , 1
1 fEach Farey sequence is a list of fractions in increasing order between 0 and 1, written in simplest form with a denominator less than or equal to the integer n. For any n greater than 1, there are an odd number of terms in the sequence and the middle term is 12.
Problem
What are the terms of the Farey sequence for n 5 5?
Th e Farey sequence for n 5 5 contains all the terms of the Farey sequence F4 plus the fractions between 0 and 1 which have a denominator of 5 when written in simplest form.
Th e fractions 05 and 55 will not be added because they simplify to 01 and 11. Insert the
fractions 15 , 25 , 3
5, and 45 in the Farey sequence F4.
F5: e 01 , 1
5 , 14 , 1
3 , 25 , 1
2 , 35 , 2
3 , 34 , 4
5 , 11 f
Exercises
1. How many terms are in each of the fi rst fi ve Farey sequences?
2. What are the terms for the Farey sequence F6?
3. What will be the new terms in the Farey sequence F7?
4. Since 11 is a prime number, how many more terms will be in the sequence F11 compared to the sequence F10?
5. Is there any limit to how large n can be?
6. Can you give examples of any other sequences?
9-2 EnrichmentArithmetic Sequences
2, 3, 5, 7, 11
e 01, 16, 15, 14, 13, 25, 12, 35, 23, 34, 45, 56, 11 f
e 17, 27, 37, 47, 57, and 67 f
No, n can be any positive integer although the computations become tedious.
Answers may vary. Sample: arithmetic, geometric, and Fibonacci
To solve word problems that involve arithmetic sequences, identify the common diff erence d, the starting value a, and the number of terms in the sequence n.
Problem
As a part-time home health care aide, you are paid a weekly salary plus a fi xed fuel fee for every patient you visit. You receive $240 in a week that you visit 1 patient. You receive $250 in a week that you visit 2 patients. How much will you receive if you visit 12 patients in 1 week?
d 5 a2 2 a1 5 250 2 240 5 10 The common difference is the difference between two consecutive terms. You receive $10 per visit.
a 5 240 Identify the starting value. You receive $240 for a week with 1 visit.
n 5 12 You want to fi nd the earnings in a week in which you visit 12 patients.
an 5 a 1 (n 2 1)d Write the formula for the nth term.
5 240 1 (12 2 1)10 Substitute.
5 240 1 110 5 350 Simplify.
You will earn $350 if you visit 12 patients in 1 week.
Exercises
7. Suppose you begin to work selling ads for a newspaper. You will be paid $50/wk plus a minimum of $7.50 for each potential customer you contact. What is the least amount of money you earn after contacting eight businesses in 1 wk?
8. A boy starts a savings account for a mountain bike. He initially deposits $15. He decides to increase each deposit by $8. How much is his 17th deposit?
9. A woman is knitting a blanket for her infant niece. Each day, she knits four more rows than the day before. She knitted seven rows on Sunday. How many rows will she knit on the following Saturday?
10. Joe started a 30-min workout program this week. He wants to increase the workout by 5 min every week. How long will his program be in the 16th week?
Th e explicit formula for the nth term of an arithmetic sequence is an 5 a 1 (n 2 1)d .
• a is the starting value and d is the common diff erence.
• n is always greater than or equal to 1.
• You can write the sequence as a, a 1 d, a 1 2d, a 1 3d, c
Problem
Find the 15th term of an arithmetic sequence whose fi rst three terms are 20, 16.5, and 13.
20 2 16.5 5 3.5 First, fi nd the common difference. The difference between 16.5 2 13 5 3.5 consecutive terms is 3.5. The sequence decreases. The common
difference is 23.5.
an 5 a 1 (n 2 1) d Use the explicit formula.
a15 5 20 1 (15 2 1)(23.5) Substitute a 5 20, n 5 15, and d 5 23.5.
5 20 1 (14)(23.5) Subtract within parentheses.
5 20 1 249 Multiply.
5 229 The 15th term is 229.
Check the answer. Write a1, a2, c, a15 down the left side of your paper. Start with a1 5 20. Subtract 3.5 and record 16.5 next to a2. Continue until you fi nd a15.
Athletics During your fi rst week of training for a marathon, you run a total of 10 miles. You increase the distance you run each week by twenty percent. How many miles do you run during your twelfth week of training?
Understanding the Problem
1. How can you write a sequence of numbers to represent this situation?
.
2. Is the sequence arithmetic, geometric, or neither?
3. What is the fi rst term of the sequence?
4. What is the common ratio of the sequence?
5. What is the problem asking you to determine?
Planning the Solution
6. Write a formula for the sequence.
Getting an Answer
7. Evaluate your formula to fi nd the number of miles you run during your twelfth week of training.
Answers may vary. Sample: Start with 10, and multiply it and each successive term
by 120% or 1.2
the 12th term of a geometric sequence that represents the number of miles you run
Write an explicit formula for each sequence. Th en generate the fi rst fi ve terms.
38. a1 5 3, r 5 22 39. a1 5 5, r 5 3 40. a1 5 21, r 5 4
41. a1 5 22, r 5 23 42. a1 5 32, r 5 20.5 43. a1 5 2187, r 5 13
44. a1 5 9, r 5 2 45. a1 5 24, r 5 4 46. a1 5 0.1, r 5 22
47. Th e deer population in an area is increasing. Th is year, the population was 1.025 times last year’s population of 2537.
a. Assuming that the population increases at the same rate for the next few years, write an explicit formula for the sequence.
b. Find the expected deer population for the fourth year of the sequence.
48. You enlarge the dimensions of a picture to 150% several times. After the fi rst increase, the picture is 1 in. wide.
a. Write an explicit formula to model the width after each increase. b. How wide is the photo after the 2nd increase? c. How wide is the photo after the 3rd increase? d. How wide is the photo after the 12th increase?
Find the missing terms of each geometric sequence. (Hint: Th e geometric mean of positive fi rst and fi fth terms is the third term. Some terms might be negative.)
19. When a pendulum swings freely, the length of its arc decreases geometrically. Find each missing arc length.
a. 20th arc is 20 in.; 22nd arc is 18.5 in. b. 8th arc is 27 mm; 10th arc is 3 mm c. 5th arc is 25 cm; 7th arc is 1 cm d. 100th arc is 18 ft; 98th arc is 2 ft
Find the missing term of each geometric sequence. It could be the geometric mean or its opposite.
1. Describe how the terms of the sequence are related.
2. For any term of the sequence, how many terms does it take before the value of the term has at least doubled?
Th e doubling period of a geometric sequence is the number of terms needed to reach a term at least twice as large as a given term. What is the doubling period for the given sequence?
3. Write the fi rst ten terms of the geometric sequence a1 5 3, r 5 1.1 to two decimal places.
4. What is the doubling period for a1 5 3? for a2 5 3.3?
Although the doubling period does not depend on which term is given, it does depend on the common ratio. For what value(s) of r is the doubling period of a geometric sequence greater than 1?
Th e idea of a doubling period applies to certain everyday situations. For example, under optimum conditions, bacteria reproduce by splitting in two. Th eir numbers increase geometrically over time. Suppose at noon on a certain day, there are 1000 bacteria in a dish. At 6 p.m. on the same day, there are 8000 bacteria.
5. If a count is taken every hour, how many terms are in the geometric sequence? What is the common ratio? What is the doubling period?
6. If a count is taken every 40 min, how many terms are in the sequence? What is the common ratio? What is the doubling period?
7. In both cases, how many hours does it take the bacteria to double?
Find the missing term of each geometric sequence. It could be the geometric mean or its opposite.
13. 5, ___, 45, c 14. 2, z z, 72, c
Find the geometric mean of 5 and 45.
!xy
!45 ? 5
!225
z z
15. 14, z z, 2 14, c 16. 175, z z, 7, c 17. 1.2, z z, 43.2, c
18. Error Analysis On a recent math test, your classmate was asked to fi nd the missing term in the geometric sequence 4, ___, 256. Her answer was 130. What error did your classmate make? What is the correct answer?
19. Th e bacteria population in a petri dish was 14 at the beginning of an experiment. After 30 min, the population was 28, and after an hour the population was 56.
a. Write an explicit defi nition to represent this sequence. b. If this pattern continues, what will be the bacteria population after 4 h?
20. A corporation earned a profi t of $420,000 in its fi rst year of operation. Over the next 10 years, the company’s CEO hopes to increase the profi t by 8% each year. If the CEO reaches her goal, what will be the company’s profi t in its seventh year, to the nearest dollar?
9-3 Practice (continued) Form K
Geometric Sequences
She found the arithmetic mean of 256 and 4 rather than the geometric mean; 32
3. Which could be the missing term of the geometric sequence 5, __, 125, . . .?
25 50 75 100
4. What could be the missing term of the geometric sequence 212, __,234, . . .?
24 26.375 3 4
5. In the explicit formula for the 9th term of the geometric sequence 1, 6, 36, . . . what number is a?
1 6 36 1,679,616
6. In each successive round of a backgammon tournament, the number of players decreases by half. If the tournament starts with 32 players, which rule could predict the number of players in the nth round?
32 5 (0.5)n 32 5 0.5r n21 an 5 15n21 an 5 (32)(0.5)n21
Short Response
7. What is the 6th term of the geometric sequence 100, 50, . . .? Show your work using the explicit formula.
9-3 Standardized Test PrepGeometric Sequences
C
H
A
H
A
I
[2] 3.125; an 5 ar n21; an 5 100Q12Rn21; a6 5 100Q12R5 5 3.125;
correct term with work shown[1] incorrect term OR correct answer, without work shown[0] incorrect answers and no work shown OR no answers given
From 2000 to 2009, your friend’s landlord has been allowed to raise her rent by the same percent each year. In 2000, her rent was $1000, and in 2003, her rent was $1092.73. What was her rent in 2009?
Step 1 Identify key information in the problem. You know that your friend’s rent was $1000 in 2000. Th is means a 5 1000. You also know that her rent in 2003 was $1092.73. Th is means that a4 5 1092.73. Her rent is raised by the same percent each year, which is the same as multiplying by a constant (e.g., a 5% increase is the same as multiplying by 1.05).
Step 2 Identify missing information.You need to fi nd the common ratio r in order to fi nd the rent in 2009, a10.
Step 3 Use the explicit formula to fi nd r.
an 5 arn 2 1 Write the explicit formula.
1092.73 5 (1000)r4 2 1 Substitute a 5 1000, a4 5 1092.73, and n 5 4.
1092.73 5 1000r3 Simplify.
1.09273 5 r3 Divide each side by 1000.
1.03 5 r Take the cube root of both sides.
Step 4 Use the value of r to fi nd the rent in 2009, a10.
an 5 arn 2 1 Write the explicit formula.
a10 5 (1000)(1.03)1021 Substitute a 5 1000, r 5 1.03, and n 5 10.
a10 5 (1000)(1.03)9 Simplify.
a10 < 1304.77 Compute. Round to the nearest hundredth.
Your friend’s rent was $1304.77 in 2009.
Exercises 22. An athlete is training for a bicycle race. She increases the amount she bikes by
the same percent each day. If she bikes 10 mi on the fi rst day, and 12.1 mi on the third day, how much will she bike on the fi fth day? By what percent does she increase the amount she bikes each day?
23. By clipping coupons and eating more meals at home, your family plans to decrease their monthly food budget by the same percent each month. If they budgeted $600 in January and $514.43 in April, how much will they budget in December?
24. From 2005 to 2009, a teen raised her babysitting rates by a fi xed percent every year. If she charged $8/h in 2005 and $10.04/h in 2007, how much did she charge in 2009? What is her percent of increase each year?
Architecture In a 20-row theater, the number of seats in a row increases by three with each successive row. Th e fi rst row has 18 seats.
a. Write an arithmetic series to represent the number of seats in the theater.
b. Find the total seating capacity of the theater.
c. Front-row tickets for a concert cost $60. After every 5 rows, the ticket price goes down by $5. What is the total amount of money generated by a full house?
1. Write the explicit formula for an arithmetic sequence.
2. What are a1 and d for the sequence that represents the number of seats in each row?
a1 5 z z d 5 z z
3. Write an explicit formula for the arithmetic sequence that represents the number of seats in each row.
4. Write an arithmetic series to represent the number of seats in the theater.
5. How can you use a graphing calculator to evaluate the series?
.
6. Find the total seating capacity of the theater.
7. Write a series for the number of seats in each set of 5 rows.
8. Use your graphing calculator to evaluate each series.
9. What are the ticket prices for each set of 5 rows?
10. What is the total amount of money generated by a full house?
an 5 a1 1 (n 2 1)d
an 5 3n 1 15
a5
n51(3n 1 15); a
10
n56(3n 1 15); a
15
n511(3n 1 15); a
20
n516(3n 1 15)
930
$46,950
120; 195; 270; 345
$60; $55; $50; $45
a20
n51
(3n 1 15)
18 3
Answers may vary. Sample: Use the sum command and the sequence command
3. 4 1 9 1 14 1 c 1 44 4. (210) 1 (225) 1 (240) 1 c 1 (285)
5. 17 1 25 1 33 1 c 1 65 6. 125 1 126 1 127 1 c 1 131
7. A bookshelf has 7 shelves of diff erent widths. Each shelf is narrower than the shelf below it. Th e bottom three shelves are 36 in., 31 in., and 26 in. wide.
a. Th e shelf widths decrease by the same amount from bottom to top. What is the width of the top shelf?
b. What is the total shelf space of all seven shelves?
Write each arithmetic series in summation notation.
8. 4 1 8 1 12 1 16 9. 10 1 7 1 4 1 c 1 (25)
10. 1 1 3 1 5 1 c 1 13 11. 3 1 7 1 11 1 c 1 31
12. (220) 1 (225) 1 (230) 1 c 1 (265) 13. 15 1 25 1 35 1 c 1 75
Find the sum of each fi nite series.
14. a4
n51(n 2 1) 15. a
6
n52(2n 2 1) 16. a
8
n53(n 1 25)
17. a5
n52(5n 1 3) 18. a
4
n51(2n 1 0.5) 19. a
6
n51(3 2 n)
20. a10
n55n 21. a
4
n51(2n 2 3) 22. a
6
n53(3n 1 2)
Use a graphing calculator to fi nd the sum of each series.
Write each arithmetic series in summation notation.
13. 3 1 8 1 13 1 c1 268
Find an explicit formula Find the value of n for 268. Write the summation notation.for the nth term. 268 5 5n 2 2
an 5 a1 1 (n 2 1)d 270 5 5n a
n51
Qz zRan 5 3 1 (n 2 1)5 54 5 n
an 5 5n 2 2
14. 1 1 7 1 13 1 c1 343 15. 5 1 7 1 9 1 c1 131
16. Tabitha used tiles to make the design shown at the right. Th e fi rst column has 2 tiles, the second column has 4 tiles, and the pattern continues.
a. Write an explicit formula for the sequence. b. Write the summation notation for a related series with 24 tiles
in the 12th column. c. How many tiles are in the design if there are a total of 12 columns?
17. Your brother is preparing for basketball season. He shot 26 baskets on the fi rst day that he practiced. He shot 32 baskets on the second day and 38 baskets the day after that.
a. If this pattern continues, how many baskets will he shoot on the 30th day? b. How many baskets will he have shot during those 30 days?
35. An embroidery pattern calls for fi ve stitches in the fi rst row and for three more stitches in each successive row. Th e 25th row, which is the last row, has 77 stitches. Find the total number of stitches in the pattern.
36. A marching band formation consists of 6 rows. Th e fi rst row has 9 musicians, the second has 11, the third has 13 and so on. How many musicians are in the last row and how many musicians are there in all?
37. Writing Explain how you can identify the diff erence between a series and a sequence.
38. a. Open-Ended Write three explicit formulas for arithmetic sequences. b. Write the fi rst seven terms of each related series. c. Use summation notation to rewrite the series. d. Evaluate each series.
39. Error Analysis A student identifi es the series 10 1 15 1 20 1 25 1 30 as an infi nite arithmetic series. Is he correct? Explain.
40. Mental Math Use mental math to evaluate a3
1(2n 1 1).
41. To train new employees, an employer off ers a bonus after 30 work days as follows. An employee must turn in one report on the fi rst day; the number of reports for each subsequent day must increase by two. What is the minimum number of reports an employee will have to turn in over the 30 days to earn the bonus?
9-4 Practice (continued) Form G
Arithmetic Series
sequence; fi nite
series; fi nite
A series is the sum of terms in a sequence, which is indicated by summation notation or addition signs.
Check students’ work. Sequences should be arithmetic and contain seven terms.
No; the series is a fi nite arithmetic series. An infi nite arithmetic series would continue indefi nitely.
3. Th e high school choir is participating in a fundraising sales contest. Th e choir will receive a bonus if they make 20 sales in their fi rst week and improve their sales by 3 in every subsequent week. What is the minimum number of sales the choir could make in the fi rst 12 weeks to qualify for the bonus?
13 53 438 5015
4. What is summation notation for the series 5 1 7 1 9 1 c 1 105?
a51
n51(2n 1 3) a
51
n51(n 1 3) a
50
n51(2n 1 3) a
51
n57(n 1 3)
5. What is the upper limit of the summation a100
n51(n 2 2)?
1 2 98 100
6. What is the sum of the series a30
n51(2n 1 2)?
62 66 990 1980
Short Response
7. What is the sum of the fi nite arithmetic series 2 1 4 1 6 1 c 1 50? Show your work.
B
H
C
F
D
H
[2] 650; Sn 5 n2(a1 1 an); S25 5 25
2 (2 1 50) 5 (12.5)(52) 5 650[1] incorrect sum OR correct sum, without work shown[0] incorrect answer and no work shown OR no answer given
Th e debate club is off ering a prize at the end of 10 weeks to a current member who brings three new members for the fi rst meeting, and then increases the number of new members they bring each week by two thereafter. One member qualifi ed for the prize with the minimum number of new members. How many new members did the member bring at Week 10? For all 10 weeks?
Step 1 Identify key information in the problem.
To win the prize, a member must bring three members to the fi rst meeting, so a 5 3.
A member must also bring two more new members to each meeting, so d 5 2.
Th e contest extends for 10 weeks, so n 5 10.
Step 2 Identify the information you are trying to fi nd.
You want to fi nd the 10th term, a10, and the sum of the fi rst 10 terms, S10.
Step 3 Use the explicit formula to fi nd a10.
an 5 a 1 (n 2 1)d Write the explicit formula. a10 5 3 1 (10 2 1)2 Substitute a 5 3, d 5 2, and n 5 10. a10 5 21 Simplify.
To win the prize, a member brought 21 new members to a meeting at Week 10.
Step 4 Use the value of a10 to fi nd the total number of new members brought by the winner.
Sn 5n2(a1 1 an) Write the formula for the sum of an arithmetic series.
S10 5102 (3 1 21) Substitute a1 5 3, a10 5 21, and n 5 10.
S10 5 120 Simplify.
Th e debate club had 120 new members brought in by the winner of the contest.
Exercises
9. Th e seating arrangement for a recital uses 20 seats in the fi rst row and two additional seats in each row thereafter. How many seats will be in the eighth row? In the ninth row? How many seats total are there in the fi rst nine rows?
10. With the help of a tutor, a student’s weekly quiz scores have increased during the fi rst four quizzes: 65, 70, 75, and 80. If the scores continue to increase at this rate, what will be the score in the 7th week? In the 8th week? What is the total of the fi rst eight scores?
If your teacher asked you to add the numbers from 1 to 100, you would probably begin by adding 1 1 2 1 3 1 c1 100, term by term from left to right. Karl Friedrich Gauss (1777–1855) found another way. Let S represent the fi nite series whose sum you are trying to fi nd. Since addition is commutative, both equations below represent this series.
S 5 001 1 92 1 93 1 c1 98 1 99 1 100
S 5 100 1 99 1 98 1 c1 93 1 92 1 001
1. What is the sum of the left side of the fi rst equation and the left side of the second equation?
2. What is the sum of each vertically-aligned pair of quantities on the right side of the equal signs?
3. How may such pairs are there?
4. Because each pair has the same sum, use multiplication to express the sum of all the pairs on the right side.
5. Write an equation that states that the sum of the left sides must equal the sum of the right sides. Solve your equation for S.
Use the technique outlined above to derive the formula for the sum of n terms of any arithmetic series. Suppose that the series starts with the term a1 and has a common diff erence of d.
6. What is the nth term, in terms of a1, d, and n?
7. Write the sum S of the n terms of the series, where each number is written in terms of a1 and d. Th en write the sum in reverse order, lining up terms.
8. What is the sum of each vertical pair of quantities on the right side?
9. How many such pairs are there?
10. Express the sum of all the pairs using multiplication.
11. Write an equation that states that the sum of the left sides must equal the sum of the right sides. Solve your equation for S.
12. Show that your equation is equivalent to S 5 n2(a1 1 an). Hint: Use your answer to Exercise 6.
2S
101
100
100 3 101 5 10,100
2S 5 10,100; S 5 5050
a1 1 (n 2 1)d
2a1 1 (n 2 1)dn
nf2a1 1 (n 2 1)dg
2S 5 nf2a1 1 (n 2 1)dg; S 5nf2a1 1 (n 2 1)dg
2
S 5nf2a1 1 (n 2 1)dg
2
5nfa1 1 a1 1 (n 2 1)dg
2
5nfa1 1 fa1 1 (n 2 1)dgg
2
5nfa1 1 ang
2 5 n2
(a1 1 an)
S 5 a1 1 (a1 1 d ) 1 c1 fa1 1 (n 2 1)d g ; S 5 fa1 1 (n 2 1)d g 1 c1 (a1 1 d ) 1 a1
7. Th is month, your friend deposits $400 to save for a vacation. She plans to deposit 10% more each successive month for the next 11 months. How much will she have saved after the 12 deposits?
Determine whether each infi nite geometric series diverges or converges. State whether each series has a sum.
8. 3 132 1
34 1 c 9. 4 1 2 1 1 1 c 10. 17 1 15.3 1 13.77 1 c
11. 6 1 11.4 1 21.66 1 c 12. 220 2 8 2 3.2 2 c 13. 50 1 70 1 98 1 c
Evaluate each infi nite geometric series.
14. 8 1 4 1 2 1 1 1 c 15. 1 113 1
19 1
127 1 . . .
16. 120 1 96 1 76.8 1 61.44 1 c 17. 1000 1 750 1 562.5 1 421.875 1 c
18. Suppose your business made a profi t of $5500 the fi rst year. If the profi t increased 20% per year, fi nd the total profi t over the fi rst 5 yr.
19. Th e end of a pendulum travels 50 cm on its fi rst swing. Each swing after the fi rst, it travels 99% as far as the preceding swing. How far will the pendulum travel before it stops?
20. A seashell has chambers that are each 0.82 times the length of the enclosing chamber. Th e outer chamber is 32 mm around. Find the total length of the shell’s spiraled chambers.
21. Th e fi rst year a toy manufacturer introduces a new toy, its sales total $495,000. Th e company expects its sales to drop 10% each succeeding year. Find the total expected sales in the fi rst 6 years. Find the total expected sales if the company off ers the toy for sale for as long as anyone buys it.
Communications Many companies use a telephone chain to notify employees of a closing due to bad weather. Suppose a company’s CEO calls three people. Th en each of these people calls three others, and so on.
a. Make a diagram to show the fi rst three stages in the telephone chain. How many calls are made at each stage?
b. Write the series that represents the total number of calls made through the fi rst six stages.
c. How many employees have been notifi ed after stage six?
1. What type of diagram can you make to represent the telephone chain?
2. Make a diagram to show the fi rst three stages in the telephone chain.
3. What expression represents the number of calls made at stage n?
4. Write the series that represents the total number of calls made through the fi rst six stages.
5. What is the sum of this series?
6. Write the sum formula.
7. Use the sum formula to fi nd how many employees have been notifi ed after stage six.
8. Does your answer agree with your sum from Exercise 5?
32. Open Ended Write an infi nite geometric series that converges to 2. Show your work.
Find the specifi ed value for each infi nite geometric series.
33. a1 5 5, S 5253 , fi nd r 34. S 5 108, r 5 13, fi nd a1
35. a1 5 3, S 5 12, fi nd r 36. S 5 840, r 5 0.5, fi nd a1
37. Error Analysis Your friend says that an infi nite geometric series cannot have a sum because it’s infi nite. You say that it is possible for an infi nite geometric series to have a sum. Who is correct? Explain.
38. Writing Describe in general terms how you would fi nd the sum of a fi nite geometric series.
9-5 Practice (continued) Form G
Geometric Series
You are; an infi nite geometric series with »r… less than 1 has a series of partial sums that converges towards a number.
Identify the fi rst term, common ratio, and nth term. Use the explicit formula to fi nd n. Then, use the sum formula with the fi rst term, common ratio, and n to fi nd the sum of the series.
Solve each exercise and enter your answer in the grid provided.
1. What is the value of a1 in the series a20
n50
3Q12R
n?
2. What is the sum of the geometric series 2 1 6 1 18 1 c1 486?
3. A community organizes a phone tree in order to alert each family of emergencies. In the fi rst stage, one person calls fi ve families. In the second stage, each of the fi ve families calls another fi ve families, and so on. How many stages need to be reached before 600 families or more are called?
4. What is the approximate whole number sum for the fi nite geometric series
a5
n50
8Q14Rn
?
5. What is the sum of the geometric series 1 113 1
An infi nite geometric series converges if the absolute value of the common ratio is less than 1 ( u r u , 1). A power series is an infi nite series where each term depends on a variable x. Each value of x will give you a specifi c infi nite series, which may converge or diverge.
1. Evaluate the expression 11 2 x for x 5 2 and for x 5
12.
2. You can evaluate many expressions with a power series called a Taylor series.
To evaluate 11 2 x , you use the Taylor series 1 1 x 1 x2 1 x3 1 c. What is
this infi nite series written in summation notation?
3. Determine the sum of the fi rst fi ve terms of the Taylor series
1 1 x 1 x2 1 x3 1 cfor x 5 2 and for x 5 12.
4. For which value of x is your computation above a better approximation for the
value of the expression 112x? What might need to be true about the value of x
in order for this Taylor series to converge to the value of this expression?
5. You can use a diff erent Taylor series to evaluate ex. Write the Taylor series
1 1x1! 1
x2
2! 1x3
3! 1 cin summation notation.
6. Evaluate ex for x 512. Evaluate the fi rst four terms of the Taylor series
1 1x1! 1
x2
2! 1x3
3! 1 cfor x 512. Round your answers to the nearest
thousandth.
7. Does the Taylor series for ex still converge if u x u $ 1? Does it give you the same value as the function y 5 ex? Explain your reasoning.
21; 2
a`
n50
xn
31; 1.9375
12;»x… R 1
a`
n50
xn
n!
1.649; 1.646
Answers may vary. Sample: Yes; yes; for any value of x, the terms eventually decrease rapidly to 0. If you add up enough terms, you will get a good approximation.
Determine whether each infi nite geometric series diverges or converges. Find the sum if the series converges.
8. 1 114 1
116 1 c 9. 2 1 8 1 32 1 c
Because ur u 5 P 14 P , 1, the series converges.
S 5a1
1 2 r 51
1 2 14
5 z z
10. 12 1
116 1
1128 1 c 11. 1
4 138 1
916 1 c 12. 2 2 25 1 2
25 2 c
13. Your classmate is trying to cut down on the amount of time he spends watching television. In January, he spent a total of 3600 min watching television. He watched television for 3240 min in February and 2916 min in March. If this pattern continues, how many minutes of television will he watch this year?
14. Your math teacher asks you to choose between two off ers. Th e fi rst off er is to receive one penny on the fi rst day, 3 pennies on the second day, 9 pennies on the third day, and so on, for 14 days. Th e second off er is to receive 4 pennies on the fi rst day, 8 pennies on the second day, 16 pennies on the third day, and so on, for 14 days. Which off er is better? What is the diff erence between the total amounts received?
6. Find the sum of the geometric series 2 2 4 1 8 2 16 1 c1 8192. Explain how you found the sum.
7. A family farm produced 2400 ears of corn in its fi rst year. For each of the next 9 yr, the farm increased its yearly corn production by 15%. How many ears of corn did the farm produce during this 10-yr period?
N 16
24092
48,729
5462; I used the explicit formula to determine that there are 13 terms in the series. Then I used the sum formula to determine the sum.
Your neighbor hosts a family reunion every year. In 2000, it costs $1500 to host the reunion. Th eir expenses have decreased by 10% per year by asking family members to contribute food and party supplies. a. What is a rule for the cost of the family reunion? b. What was the cost of the reunion in 2005? c. What was the total cost for hosting the family reunions from 2000 to 2009?
Th e cost is a geometric sequence that decreases by the same percent each year.
an 5 arn21 Write the explicit formula.
an 5 (1500)(0.90)n21 Substitute a1 5 1500, r 5 1 2 0.10 5 0.90 in the explicit formula.
To fi nd the cost of the reunion in 2005 (n 5 6), substitute values into the explicit formula.
an 5 arn21 Write the explicit formula.
an 5 (1500)(0.90)621 Substitute a1 5 1500, r 5 0.90, n 5 6 in the formula.
an < 886 Simplify.
Th e cost of hosting the reunion in 2005 was $886.
To fi nd the total of hosting the reunions from 2000 to 2009, a10
n51(1500)(0.90)n21,
fi nd the sum of the geometric series.
Sn 5a1(1 2 rn)
1 2 r Write the formula for the sum of a geometric series.
S10 51500(1 2 0.9010)
1 2 0.90 Substitute a1 5 1500, r 5 0.90, n 5 10 in the formula.
S10 < 9770 Simplify.
Th e cost of hosting the reunions from 2000 to 2009 was $9770.
Exercise
14. In 1990, a vacation package cost $400. Th e cost has increased 10% per year. a. What are the values of a1 and r? b. What is a rule for the cost of the vacation? c. What was cost of the vacation in 1995? d. What was the total cost of the vacations from 1990 to 1999? e. If the pattern continued until 2009, what was the total cost of the vacations?
Answers may vary. Sample: No, the terms of the series grow in absolute value so they cannot have a fi nite sum; two possible series with r 5 2 are 3 1 6 1 12 1 24 1 . . . and (21) 1 (22) 1 (24) 1 . . . .
4. 80, 40, 20, 10, c 5. 4, 10, 16, 22, c 6. 3, 21, 147, 1029, c
Find the 18th term of each arithmetic sequence.
7. 5, 9, 13, 17, c 8. 4, 1, 22, 25, c 9. 1.2, 1.6, 2, 2.4, c
Use the arithmetic mean to fi nd the missing term in each arithmetic sequence.
10. c6, z z, 28, c 11. c2, z z, 214, c 12. c1.4, z z, 6.8, c
Do you UNDERSTAND?
13. Writing Describe the diff erence between a recursive defi nition and an explicit defi nition of a sequence.
14. Tim takes the stairs up to his offi ce. He enters the ground fl oor of the building and climbs 12 steps to reach the fi rst fl oor. He climbs a total of 24 steps to reach the second fl oor and 36 steps to reach the third fl oor. How many steps will Tim climb to reach his offi ce on the 16th fl oor?
3, 8, 13, 18
a1 5 80; an11 5 12an
73
17
An explicit defi nition describes the nth term of a sequence using the number n.A recursive defi nition relates each term to the next.
1. 4, 12, 36, 108, c 2. 2, 1, 12, 14, c 3. 0.04, 0.2, 1, 5, c
Find the sum of each fi nite arithmetic series.
4. 3 1 6 1 9 1 c1 72 5. 6 1 12 1 18 1 c1 108
6. (22) 1 (27) 1 (212) 1 c1 (2102)
Write each arithmetic series in summation notation.
7. 1 1 5 1 9 1 c1 85 8. 5 1 11 1 17 1 c1 371
9. 212 1 204 1 196 1 c1 (220)
Find the sum of each fi nite geometric series.
10. 5 1 15 1 45 1 c1 10,935 11. 16 1
112 1
124 1 c1 1
384
12. 1 2 4 1 16 2 c2 16,384
Do you UNDERSTAND?
13. A guitar-making company produced 60 guitars this month. Th e company plans to increase production by 8% each month for the next 9 months. How many guitars will they produce during this 10-month period?
34. On October 1, a gardener plants 20 bulbs. On October 2, she plants 23 bulbs. On October 3, she plants 26 bulbs. She continues in this pattern until October 15, when she plants the last bulbs.
a. Write an explicit formula to model the number of bulbs she plants each day. b. Write a recursive defi nition to model the number of bulbs she plants each day. c. How many bulbs will the gardener plant on October 15? d. What is the total number of bulbs she plants from October 1 to October 15, inclusive?
35. Suppose you are building 10 steps with 6 concrete blocks in the top step and 60 blocks in the bottom step. If the number of blocks in each step forms an arithmetic sequence, fi nd the total number of concrete blocks needed to build the steps.
Do you UNDERSTAND?
36. Writing Explain why an infi nite geometric series with r 5 1 diverges. Include an example in your explanation.
37. Open-Ended Write a sequence and describe it using both an explicit defi nition and a recursive formula.
38. Reasoning What does a recursive defi nition have that an explicit formula does not? Explain.
12.5120
24
330 blocks
210
83
28 1
10
arithmetic; 230
an 5 20 1 3(n 2 1)an 5 an21 1 3 where a1 5 2062 bulbs
because the number 5 is added an infi nite number of times
Answers may vary. Sample: 30; 300; 3000; 30,000; 300,000; . . .; an 5 10an21 where a1 5 30; an 5 30(10)n21
Answers may vary. Sample: A recursive defi nition contains an initial condition as well as a formula for how to move from one term to the other. An explicit formula describes the nth term in terms of n.
22. Error Analysis Your friend calculated the sum of the fi nite geometric series 2 1 8 1 32 1 c1 32,768. Her answer was 131,080. What error did she make? What is the correct sum?
23. Writing Find the possible values of the missing term in the following geometric sequence, and explain how you found the answer.
6, z z, 96, c
Chapter 9 Test (continued) Form K
1280
1736
3160
She used the formula for the sum of an arithmetic series rather than the sum of a geometric series; 43,690
Answers may vary. Sample: First, I found the product of 96 and 6, which is 576. Then I found the square root of 576, which is ±24.
Write an explicit formula for each sequence. Th en fi nd the 12th term.
4. 2, 6, 12, 20, c 5. 2.5, 3, 3.5, 4, c 6. 2, 5, 10, 17, 26, c
Find the 20th term of each arithmetic sequence.
7. 2, 5, 8, 11, c 8. 56, 50, 44, 38, c 9. 2.2, 2.6, 3, 3.4, c
Do you UNDERSTAND?
10. Writing Find the missing term in the sequence below. Th en explain how you found the term.
c12, z z, 44, c
11. Reasoning Rita must fi nd the 35th term in the sequence that begins 2, 9, 16, 23, c. She needs to fi nd the answer as fast as possible. Should Rita use a recursive defi nition or an explicit formula? Why?
12. A bus has 6 people on it as it pulls out of the station to begin its route. After one stop, there are 11 people on the bus. After the second stop, there are 16 people on the bus. If this pattern continues, how many people will be on the bus after 10 stops?
Chapter 9 Test Form K
7, 10, 13, 16, 19
an 5 n(n 1 1); 156
59
51 people
Answers may vary. Sample: First, I found the sum of 12 and 44, which is 56. Then I divided the sum by 2 to fi nd the missing term, 28.
recursive defi nition will require her to go through many iterations of the defi nition. If she uses an explicit formula, she will be able to substitute the value into the formula to fi nd the answer.
Task 1 a. Use your graphing calculator to graph the function f(x) 5 2x over the domain
5x | x $ 06 . b. Use the TABLE feature on your calculator to make a table of values of the
function f for the set of x-values 1, 2, 3, . . . . c. Determine whether the sequence of function values is arithmetic, geometric,
or neither. Justify your response. d. Write a recursive defi nition and an explicit formula for the sequence of function
values. e. Find three terms of the sequence between 512 and 8192, and identify these
as arithmetic or geometric means. Explain your reasoning.
Task 2 a. Determine whether the sequence 27, 9, 3, 1, . . . is geometric, arithmetic, or neither.
Justify your response. b. Write a recursive defi nition and an explicit formula for this sequence. c. Use summation notation to write the series related to the fi rst ten terms
of the sequence give in part (a). Th en evaluate this series. d. Use summation notation to write the series related to the infi nite
sequence given in part (a). Determine whether this series diverges or coverages. If the series converages, fi nd its sum.
e. Describe a real-world situation that can be modeled by this sequence.
[4] Student correctly uses calculator to view the function and constructs table of values using positive integers for x-values. Student correctly identifi es the sequence as geometric and justifi es answer. Student correctly writes recursive defi nition and explicit formula for the sequence, fi nds three terms, and identifi es these as geometric means with justifi cation.
[3] Student correctly completes parts (a), (b), and (c). Justifi cation may not be fully developed. Student completes parts (d) and (e) with only minor errors and some justifi cation.
[2] Student correctly completes parts (a), (b), and (c). Justifi cation is not given. Student writes recursive defi nition and explicit formula with one or more errors. Student fi nds three terms with one or more major errors. Justifi cation is not given.
[1] Student determines minimal and/or incorrect information about the sequence, its formulas, and the geometric means. There are major errors in logic.
[0] Student makes no attempt, or no response is given.
1024, 2048, and 4096; geometric means; 2048 is the square root of the product of 512 and 8192, 1024 is the square root of the product of 512 and 2048, and 4096 is the square root of the product of 2048 and 8192.
an 5 2an21 where a1 5 2; an 5 2n
geometric; There is a common ratio of 2.
Y1 5 2ˆX
Y 5 32X 5 5
X248163264
123456X 5 0
Y1
geometric; There is a common ratio of 13. an 5 13 an21 where
a1 5 27; an 5 27Q13Rn21
a10
n5127Q13Rn21; 29,524
729
a`
n5127Q13Rn21; converges; 81
2
[4] Student correctly determines that the sequence is geometric. Student correctly fi nds a recursive defi nition and explicit formula for the sequence. Student correctly writes the series using summation notation, fi nds the sum of the fi rst ten terms of the series, determines that the infi nite series converges, and correctly determines the sum. Student describes a feasible real-world situation.
[3] Student completes all parts with only minor errors. Student describes a feasible real-world situation.
[2] Student completes all parts with one or more major errors.[1] Student determines minimal and/or incorrect information about the sequence and
series, their recursive defi nitions and explicit formulas, and the series in summation notation. There are major errors in logic.
[0] Student makes no attempt, or no response is given.
Task 3 a. Determine whether the sequence 2, 8, 14, 20, 26,c is arithmetic geometric, or neither.
Justify your response. b. Write a recursive defi nition and an explicit formula for this sequence. c. Find three terms of the sequence between 62 and 86, and identify
these as arithmetic or geometric means. Explain your reasoning.
d. Use summation notation to write the series related to the infi nite sequence given in part (a). Find the sum of the fi rst ten terms of the series.
e. Describe a real-world situation that can be modeled by the sequence given in part (a).
Task 4 a. Graph the function f (x) 5 20.5x2 1 4.5 for the domain 23 # x # 3 using
your graphing calculator. b. Carefully draw the graph of the function on a sheet of graph
paper. c. Draw and use inscribed rectangles 1 unit wide to approximate the area
under the curve for the given interval. d. Use e f (x)dx feature from the CALC menu of your graphing calculator
to determine the area under the curve for the given interval.
Arithmetic; there is a common difference of 6.
Check student’s drawing.
13 units2
18 units2
Check students’ work.
an 5 6 1 an21 where a1 5 2; an 5 2 1 6(n 2 1)
68, 74, and 80; arithmeticmeans; 74 is the average of 62
and 86, 68 is the average of 62 and 74, and 80 is the average of 74 and 86.
a`
n51(6n 2 4); 290
[4] Student uses a calculator to view the function and constructs a table of values using positive integers for x-values. Student correctly identifi es the sequence as geometric and justifi es answer. Student writes a recursive defi nition and an explicit formula for the sequence, fi nds three terms, and identifi es these as arithmetic means with justifi cation. Student uses summation notation to write a series and correctly fi nds the sum of the fi rst ten terms. Student describes a real-world situation.
[3] Student completes all parts with minor errors.[2] Student makes major errors in one or more parts.[1] Student determines minimal and/or incorrect information about the sequence, its
formulas, and the geometric means. There are major errors in logic.[0] Student makes no attempt, or no response is given.
10
25
23 3
[4] Student correctly graphs the function over the designated domain on a graphing calculator. Student draws a neat graph of the function on graph paper. Student makes a close estimate of the area under the curve using rectangles. Student correctly uses a graphing calculator to fi nd the area under the curve.
[3] Student completes all parts with only minor errors.[2] Student makes major errors in one or more parts.[1] Student determines minimal and/or incorrect information about the graph and the area
under it. There are major errors in logic.[0] Student makes no attempt, or no response is given.
About the ProjectTh e Chapter Project gives students an opportunity to use sequences, explicit formulas, and recursive formulas to change the sizes of drawings and photos. Th ey investigate perspective, the use of grids to enlarge and reduce, and ways to crop, enlarge, and reduce photographs.
Introducing the Project• Ask students if they have ever seen artists draw buildings or other objects that
appear to recede in the distance.
• Ask them why it appears that railroad track rails get closer together when we look at them in the distance.
• Explain that they will investigate the concepts of perspective and vanishing points, and the mathematics involved in enlarging, reducing, and cropping pictures and photographs.
Activity 1: ResearchingStudents research perspective, create drawings in perspective, write arithmetic sequences, and determine explicit or recursive formulas for their sequences.
Activity 2: DesigningStudents use grid paper to enlarge designs. Th ey use the same ratios repeatedly to draw lengths which form geometric sequences. Th ey then write explicit or recursive formulas for their sequences.
Activity 3: AnalyzingStudents crop photos. Th en they enlarge the cropped portions, writing sequences for the widths of the enlargements.
Finishing the ProjectYou may wish to plan a project day on which students share their completed projects. Encourage students to explain their processes as well as their results.
• Have students review their methods for writing explicit and recursive formulas for arithmetic and geometric sequences.
• Ask groups to share their insights that resulted from completing the project, such as any shortcuts they found for making their drawings or writing formulas.
Short Response 9. Graph the system of inequalities e y , 2x 2 1
y $ 2x 1 3.
10. Describe how the graph of y 5 log 3(x 2 2) 1 5 compares to the graph of the parent function.
11. How can the relationship between variables in the table be described?
12. Use the sequence 100, 95, 90, 85, . . . a. Describe the sequence in words. b. Find the next three terms.
13. Water leaks from a 10,000-gal tank at a rate of 5 gal/h. Write a linear model for the situation and use it to fi nd the amount of water in the tank after 24 h.
Extended Response
14. You have a coupon for $10 off a CD. You also get a 20% discount if you show your membership card in the CD club. How much more would you pay if the cashier applies the coupon fi rst? Use composite functions. Show your work.
x
1
2
4
5
y
20
10
5
4
The graph of y 5 log3(x 2 2) 1 5 is a shift of the graph of the parent function y 5 log3x to the right two units and up fi ve units.
w 5 25t 1 10,000; 9880 gal
[4] $2; student defi nes both functions and subtracts one from the other correctly.[3] Student defi nes both functions and subtracts one from the other with minor errors.[2] Student determines minimal and/or incorrect information about the functions and
does not subtract one from the other. There are major errors in logic.[1] Student provides incorrect information. No work is shown.[0] Student makes no attempt or no response is given.
The variables in the table, x and y, have an inverse variation relationship. As x-values increase, y-values decrease, but the product of their pairs remains constant.
42 6 8�2�4
2468
�4
x
O
y
This is an arithmetic sequence in which each term is fi ve less than the previous one.
Beginning the Chapter ProjectWhen a book is being made, artists, designers, and photographers work with writers and editors to make the pages visually attractive. Th ese professionals often work with patterns involving arithmetic and geometric sequences.
In this project, you will see how perspective aff ects perceived lengths and distances. You will use grids to change the sizes of drawings. You also will learn how a designer crops a photo, then enlarges or reduces it.
ActivitiesActivity 1: ResearchingResearch the concepts of one- and two-point perspective and vanishing points in art.
• Measure the lengths of the arrows shown at the right. What is the relationship between these lengths? How does this relate to your research on perspective?
• Trace the four arrows at the right, moving the paper to the left after tracing the longest arrow so that it is further away from the others than it is now. What do you notice?
• Make a simple drawing of three or more similar objects whose lengths can be represented by an arithmetic sequence. Write the corresponding arithmetic sequence, and a recursive or explicit formula for that sequence.
Activity 2: DesigningWhen a book is made, a designer or artist may change the size of an original sketch to fi t the space available on a page. One way to change the dimensions of a sketch is to use graph paper with diff erent size squares.
• Draw a fi gure or design on a sheet of graph paper. Label this Figure 1 and record its approximate dimensions.
• Enlarge the original fi gure by copying each portion of Figure 1, square by square, onto larger squares. Label this Figure 2 and record its dimensions.
• Use a ratio to compare the dimensions of Figure 1 to the dimensions of Figure 2. If the same ratio is used to enlarge Figure 2, what would the dimensions of the new fi gure be? Draw this fi gure, label it Figure 3, and record its dimensions.
• Explain why the lengths of the three fi gures form a geometric sequence.• Write a geometric sequence corresponding to these lengths, and a recursive or explicit
formula for that sequence.
Figure 1 Figure 2
• Check students’ work; answers may vary. Sample: The lengths form an arithmetic sequence; answers may vary. Sample: The lines of sight along the tops and bottoms of the arrows meet at a vanishing point.
• Check students’ work; answers may vary. Sample: There is no longer a vanishing point.• Check students’ work.
Getting StartedRead the project. As you work on the project, you will need a calculator, a metric ruler, at least two types of graph paper, and materials on which you can record your calculations. Keep all of your work for the project in a folder.
Checklist Suggestions
☐ Activity 1: relating perspective and arithmetic sequences
☐ Use art books from the school library or the Internet.
☐ Activity 2: relating dimensions and geometric sequences
☐ Use grid paper to draw simple geometric designs.
☐ Activity 3: relating photo-cropping and sequences
☐ Measure directly or use proportions to fi nd the widths.
☐ presentation ☐ Does your display include examples of both arithmetic and geometric sequences? What artists or work of art with which you are familiar best demonstrate the concepts of one-point perspective, two-point perspective, or vanishing points?
Scoring Rubric4 Calculations, sequences, and formulas are correct. Drawings are neat,
accurate, and clearly show the sequences. Explanations are thorough and well thought out.
3 Calculations, sequences, and formulas are mostly correct with some minor errors. Drawings are neat and mostly accurate. Explanations lack detail or are not completely accurate.
2 Calculations contain both minor and major errors. Drawings are not accurate.
1 Major concepts are misunderstood. Project satisfi es few of the requirements and shows poor organization and eff ort.
0 Major elements of the project are incomplete or missing.
Your Evaluation of Project Evaluate your work, based on the Scoring Rubric.
Activity 3: AnalyzingPhotographs are often cropped so that only part of the photograph remains. Th en, this cropped portion can be reduced or enlarged. Choose a photograph in a textbook. Place a piece of paper over the photograph, trace its original size, and draw a rectangle to indicate a portion of the photograph that you would like to crop. Draw a diagonal from the lower left corner to the upper right corner of the rectangular cropped area. If this diagonal is extended through the upper right corner of the cropped area, and a point selected anywhere along the diagonal or its extension, then the rectangle having the chosen point as its upper right corner (and the same lower left corner as the original cropped area) will have dimensions that are proportional to the dimensions of the cropped area.
• Measure the dimensions and the length of the diagonal of the cropped area.
• Write the fi rst four terms of an arithmetic sequence that has the length of the diagonal of the cropped area as its fi rst term. Using the terms of your sequence as diagonal lengths, fi nd the four corresponding photo widths. What do you notice about this list of widths?
• Write the fi rst four terms of an geometric sequence that has the length of the diagonal of the cropped area as its fi rst term. Using the terms of your sequence as diagonal lengths, fi nd the four corresponding photo widths. What do you notice about this list of widths?
Finishing the ProjectTh e answers to the activities should help you complete your project. Prepare a presentation or demonstration that summarizes how an artist, a designer, or a photographer uses sequences. Present this information to your classmates. Th en discuss the sequences you made.
Refl ect and ReviseReview your summary. Are your drawings clear and correct? Are your sequences accurate? Practice your presentation in front of at least two people before presenting it to the class. Ask for their suggestions for improvement.
Extending the ProjectGeometric and arithmetic patterns are used in other aspects of design and in other careers. Research other areas where sequences are applied.