sequences series

TRIGONOMETRY Name: ___________________________________ Unit 6: Sequences & Series Period: _________

Flipped Classroom

In your final unit of trigonometry, you will learn the material in a slightly different manner, on your own and at home. One day, youll come across a class where perhaps you have to do this merely because you do not understand your professor or perhaps you prefer not to make the trek to campus in the frigid arctic temperatures. To prepare you for this, you will complete the final unit of trigonometry by learning at home (your homework) and practicing at school in front of your teacher.

How will you be graded during this unit? Well, of course you will have a test at the end of the unit (per usual) but to ensure that you are taking notes and completing the requirements of your at home classroom, you will be graded when you arrive to class each day based upon the following: Quality of the notes from the Lesson Videos Quality of the guided note pages (show your work on the practice

problems!) Quality of any other requirements of the at home lesson Quality and Quantity of Comments/Questions/Answers made in Unit 6

Discussion Forum Effort put forth to Comprehend/Learn/Understand the material

**The Flipped Classroom Unit will count as your project for the unit** Total: 50 points

Learning Target: I can 1. Find the nth term of an arithmetic or geometric sequence 2. Find the position of a given term of an arithmetic or geometric sequence 3. Find sums of a finite arithmetic or geometric series 4. Use sequences and series to solve real-world problems 5. Use sigma notation to express sums

Resources: Guided Notes (Arithmetic Sequences/Series & Geometric Sequences/Series) Lecture Notes & Solutions (on taskstream) ASSISTment Lessons (containing instructional videos) ACT Quality Core Formula Sheet

Important Links: LT1/2 #1: http://screencast.com/t/l5nPmLtOVi LT1/2 #2: http://screencast.com/t/HTGVvCTMx LT1/2 #3: http://screencast.com/t/JbbA6TZDT LT3/5 #1:http://screencast.com/t/1M1Zob1Hfp LT3/5 #2: http://screencast.com/t/3sFFWGU9au LT3/5 #3: http://screencast.com/t/Sgc4Py0Qcgue

Unit 6 Instructional Plan:

Learning Targets 1 & 2 1. Lesson at home 5/13

Watch the three lesson videos (Complete Lecture Note Examples throughout and Check Answers on Taskstream)

Complete Guided Notes Pages 631 / 635 / 643 / 647 Make at least 2 Comments on discussion board. Your comments can be a

detailed explanation of what you learned / a new method of solving a specific problem / a question about a specific problem / an answer to someones question

2. Practice at school 5/14 Speed Dating / Sequences Taskcards P. 634 / 646 Sequences Skillbuilder

Learning Targets 3 & 5 1. Lesson at home 5/14

Watch the three lesson videos (Complete Lecture Note Examples throughout and Check Answers on Taskstream)

Complete Guided Notes Pages 637 / 638 / 641 / 649 / 650 / 653 Make at least 2 Comments on discussion board. Your comments can be a


2. Practice at school 5/15 Speed Dating / Series Taskcards P. 640 / 652 ASSISTment LT3/5

Learning Target 4 1. Lesson/Research at home 5/15

Read through (and take notes) the LT4 Lecture Notes [on Taskstream] Make at least 2 Comments on discussion board. Your comments can be a


Develop 2 of your own Real World Problems. You must do a problem of both an arithmetic and a geometric sequence or series (one problem should be a sequence, one should be a series). Clearly and concisely phrase your question and write your question in bold letters on two separate sheets of paper. You may use the example problems in the notes to help you model your question.

2. Practice at school 5/16 Showing all your work, solve the 2 problems that you created. Post your 2 questions in the room. Solve 4 other problems on a post it note.

Post your solution next to the question. Check the work of your classmates and provide feedback (think about the

types of questions that Miss Rudolph asks you never tells you exactly how!) ASSISTment LT4

Review 5/19 Unit 6 Test 5/21

Study Guide and InterventionArithmetic Sequences

NAME ______________________________________________ DATE ____________ PERIOD _____

11-111-1

Glencoe/McGraw-Hill 631 Glencoe Algebra 2

Less

on

11-

1

Arithmetic Sequences An arithmetic sequence is a sequence of numbers in which eachterm after the first term is found by adding the common difference to the preceding term.

nth Term of an an ! a1 " (n # 1)d, where a1 is the first term, d is the common difference, Arithmetic Sequence and n is any positive integer

Find the next fourterms of the arithmetic sequence 7, 11, 15, .Find the common difference by subtractingtwo consecutive terms.

11 # 7 ! 4 and 15 # 11 ! 4, so d ! 4.

Now add 4 to the third term of the sequence,and then continue adding 4 until the fourterms are found. The next four terms of thesequence are 19, 23, 27, and 31.

Find the thirteenth termof the arithmetic sequence with a1 ! 21and d ! "6.Use the formula for the nth term of anarithmetic sequence with a1 ! 21, n ! 13,and d ! #6.an ! a1 " (n # 1)d Formula for nth term

a13 ! 21 " (13 # 1)(#6) n ! 13, a1 ! 21, d ! #6a13 ! #51 Simplify.

The thirteenth term is #51.

Example 1Example 1 Example 2Example 2

Example 3Example 3 Write an equation for the nth term of the arithmetic sequence "14, "5, 4, 13, .In this sequence a1 ! #14 and d ! 9. Use the formula for an to write an equation.an ! a1 " (n # 1)d Formula for the nth term! #14 " (n # 1)9 a1 ! #14, d ! 9! #14 " 9n # 9 Distributive Property! 9n # 23 Simplify.

Find the next four terms of each arithmetic sequence.

1. 106, 111, 116, 2. #28, #31, #34, 3. 207, 194, 181, 121, 126, 131, 136 "37, "40, "43, "46 168, 155, 142, 129

Find the first five terms of each arithmetic sequence described.

4. a1 ! 101, d ! 9 5. a1 ! #60, d ! 4 6. a1 ! 210, d ! #40101, 110, 119, 128, 137 "60, "56, "52, "48, "44 210, 170, 130, 90, 50

Find the indicated term of each arithmetic sequence.

7. a1 ! 4, d ! 6, n ! 14 82 8. a1 ! #4, d ! #2, n ! 12 "269. a1 ! 80, d ! #8, n ! 21 "80 10. a10 for 0, #3, #6, #9, "27

Write an equation for the nth term of each arithmetic sequence.

11. 18, 25, 32, 39, 12. #110, #85, #60, #35, 13. 6.2, 8.1, 10.0, 11.9, 7n # 11 25n " 135 1.9n # 4.3

ExercisesExercises


Find the next four terms of each arithmetic sequence.

1. 5, 8, 11, 14, 17, 20, 23 2. #4, #6, #8, "10, "12, "14, "163. 100, 93, 86, 79, 72, 65, 58 4. #24, #19, #14, "9, "4, 1, 65. , 6, , 11, , 16, , 21 6. 4.8, 4.1, 3.4, 2.7, 2, 1.3, 0.6

Find the first five terms of each arithmetic sequence described.

7. a1 ! 7, d ! 7 8. a1 ! #8, d ! 2

7, 14, 21, 28, 35 "8, "6, "4, "2, 09. a1 ! #12, d ! #4 10. a1 ! , d !

"12, "16, "20, "24, "28 , 1, , 2, 11. a1 ! # , d ! # 12. a1 ! 10.2, d ! #5.8

" , " , " , " , " 10.2, 4.4, "1.4, "7.2, "13

Find the indicated term of each arithmetic sequence.

13. a1 ! 5, d ! 3, n ! 10 32 14. a1 ! 9, d ! 3, n ! 29 9315. a18 for #6, #7, #8, . "23 16. a37 for 124, 119, 114, . "5617. a1 ! , d ! # , n ! 10 " 18. a1 ! 14.25, d ! 0.15, n ! 31 18.75

Complete the statement for each arithmetic sequence.

19. 166 is the th term of 30, 34, 38, 35 20. 2 is the th term of , , 1, 8

Write an equation for the nth term of each arithmetic sequence.

21. #5, #3, #1, 1, an ! 2n " 7 22. #8, #11, #14, #17, an ! "3n " 523. 1, #1, #3, #5, an ! "2n # 3 24. #5, 3, 11, 19, an ! 8n " 13

Find the arithmetic means in each sequence.

25. #5, , , , 11 "1, 3, 7 26. 82, , , , 18 66, 50, 3427. EDUCATION Trevor Koba has opened an English Language School in Isehara, Japan.

He began with 26 students. If he enrolls 3 new students each week, in how many weekswill he have 101 students? 26 wk

28. SALARIES Yolanda interviewed for a job that promised her a starting salary of $32,000with a $1250 raise at the end of each year. What will her salary be during her sixth yearif she accepts the job? $38,250

??????

4$5

3$5

??

18$3$5

9$5

13$

11$

3$

7$

5$

1$3

5$6

5$

3$

1$

1$2

1$2

37$

27$17$2

7$2

Practice (Average)Arithmetic Sequences

NAME ______________________________________________ DATE ____________ PERIOD _____

11-111-1

Jessica Rudolph

Reading to Learn MathematicsArithmetic Sequences

NAME ______________________________________________ DATE ____________ PERIOD _____

11-111-1


Less

on

11-

1

Pre-Activity How are arithmetic sequences related to roofing?Read the introduction to Lesson 11-1 at the top of page 578 in your textbook.

Describe how you would find the number of shingles needed for the fifteenthrow. (Do not actually calculate this number.) Explain why your method willgive the correct answer. Sample answer: Add 3 times 14 to 2. Thisworks because the first row has 2 shingles and 3 more areadded 14 times to go from the first row to the fifteenth row.

Reading the Lesson

1. Consider the formula an ! a1 " (n # 1)d.

a. What is this formula used to find?a particular term of an arithmetic sequence

b. What do each of the following represent?

an: the nth terma1: the first term n: a positive integer that indicates which term you are findingd: the common difference

2. Consider the equation an ! #3n " 5.

a. What does this equation represent? Sample answer: It gives the nth term ofan arithmetic sequence with first term 2 and common difference "3.

b. Is the graph of this equation a straight line? Explain your answer. Sampleanswer: No; the graph is a set of points that fall on a line, but thepoints do not fill the line.

c. The functions represented by the equations an ! #3n " 5 and f(x) ! #3x " 5 arealike in that they have the same formula. How are they different? Sampleanswer: They have different domains. The domain of the first functionis the set of positive integers. The domain of the second function isthe set of all real numbers.

Helping You Remember3. A good way to remember something is to explain it to someone else. Suppose that your

classmate Shala has trouble remembering the formula an ! a1 " (n # 1)d correctly. Shethinks that the formula should be an ! a1 " nd. How would you explain to her that sheshould use (n # 1)d rather than nd in the formula? Sample answer: Each termafter the first in an arithmetic sequence is found by adding d to theprevious term. You would add d once to get to the second term, twice toget to the third term, and so on. So d is added n " 1 times, not n times,to get the nth term.

Study Guide and InterventionArithmetic Series

NAME ______________________________________________ DATE ____________ PERIOD _____

11-211-2


Less

on

11-

2

Arithmetic Series An arithmetic series is the sum of consecutive terms of anarithmetic sequence.

Sum of an The sum Sn of the first n terms of an arithmetic series is given by the formulaArithmetic Series Sn ! $

n2$[2a1 " (n # 1)d ] or Sn ! $

n2$(a1 " an)

Find Sn for thearithmetic series with a1 ! 14,an ! 101, and n ! 30.Use the sum formula for an arithmeticseries.

Sn ! (a1 " an) Sum formula

S30 ! (14 " 101) n ! 30, a1 ! 14, an ! 101

! 15(115) Simplify.! 1725 Multiply.

The sum of the series is 1725.

30$2

n$2

Find the sum of allpositive odd integers less than 180.The series is 1 " 3 " 5 " " 179.Find n using the formula for the nth term ofan arithmetic sequence.

an ! a1 " (n # 1)d Formula for nth term179 ! 1 " (n # 1)2 an ! 179, a1 ! 1, d ! 2179 ! 2n # 1 Simplify.180 ! 2n Add 1 to each side.

n ! 90 Divide each side by 2.

Then use the sum formula for an arithmeticseries.


S90 ! (1 " 179) n ! 90, a1 ! 1, an ! 179


The sum of all positive odd integers lessthan 180 is 8100.

90$2

n$2


ExercisesExercises

Find Sn for each arithmetic series described.

1. a1 ! 12, an ! 100, 2. a1 ! 50, an ! #50, 3. a1 ! 60, an ! #136,n ! 12 672 n ! 15 0 n ! 50 "1900

4. a1 ! 20, d ! 4, 5. a1 ! 180, d ! #8, 6. a1 ! #8, d ! #7,an ! 112 1584 an ! 68 1860 an ! #71 "395

7. a1 ! 42, n ! 8, d ! 6 8. a1 ! 4, n ! 20, d ! 2 9. a1 ! 32, n ! 27, d ! 3

504 555 1917Find the sum of each arithmetic series.

10. 8 " 6 " 4 " " #10 "10 11. 16 " 22 " 28 " " 112 108812. #45 " (#41) " (#37) " " 35 "105Find the first three terms of each arithmetic series described.

13. a1 ! 12, an ! 174, 14. a1 ! 80, an ! #115, 15. a1 ! 6.2, an ! 12.6,Sn ! 1767 12, 21, 30 Sn ! #245 80, 65, 50 Sn ! 84.6 6.2, 7.0, 7.8

1$2


Sigma Notation A shorthand notation for representing a series makes use of the Greek letter . The sigma notation for the series 6 " 12 " 18 " 24 " 30 is !

5

n!16n.

Evaluate !18

k!1(3k # 4).

The sum is an arithmetic series with common difference 3. Substituting k ! 1 and k ! 18into the expression 3k " 4 gives a1 ! 3(1) " 4 ! 7 and a18 ! 3(18) " 4 ! 58. There are 18 terms in the series, so n ! 18. Use the formula for the sum of an arithmetic series.


S18 ! (7 " 58) n ! 18, a1 ! 7, an ! 58


So !18

k!1(3k " 4) ! 585.

Find the sum of each arithmetic series.

1. !20

n!1(2n " 1) 2. !

25

n!5(x # 1) 3. !

18

k!1(2k # 7)

440 294 216

4. !75

r!10(2r # 200) 5. !

15

x!1(6x " 3) 6. !

50

t!1(500 # 6t)

"7590 765 17,350

7. !80

k!1(100 # k) 8. !

85

n!20(n # 100) 9. !

200

s!13s

4760 "3135 60,300

10. !28

m!14(2m # 50) 11. !

36

p!1(5p # 20) 12. !

32

j!12(25 # 2j)

"120 2610 "399

13. !42

n!18(4n # 9) 14. !

50

n!20(3n " 4) 15. !

44

j!5(7j # 3)

2775 3379 6740

18$2

n$2

Study Guide and Intervention (continued)Arithmetic Series

NAME ______________________________________________ DATE ____________ PERIOD _____

11-211-2

ExampleExample

ExercisesExercises


Find Sn for each arithmetic series described.

1. a1 ! 16, an ! 98, n ! 13 741 2. a1 ! 3, an ! 36, n ! 12 234

3. a1 ! #5, an ! #26, n ! 8 "124 4. a1 ! 5, n ! 10, an ! #13 "40

5. a1 ! 6, n ! 15, an ! #22 "120 6. a1 ! #20, n ! 25, an ! 148 1600

7. a1 ! 13, d ! #6, n ! 21 "987 8. a1 ! 5, d ! 4, n ! 11 275

9. a1 ! 5, d ! 2, an ! 33 285 10. a1 ! #121, d ! 3, an ! 5 "2494

11. d ! 0.4, n ! 10, an ! 3.8 20 12. d ! # , n ! 16, an ! 44 784

Find the sum of each arithmetic series.

13. 5 " 7 " 9 " 11 " " 27 192 14. #4 " 1 " 6 " 11 " " 91 87015. 13 " 20 " 27 " " 272 5415 16. 89 " 86 " 83 " 80 " " 20 1308

17. !4

n!1(1 # 2n) "16 18. !

6

j!1(5 " 3n) 93 19. !

5

n!1(9 # 4n) "15

20. !10

k!4(2k " 1) 105 21. !

8

n!3(5n # 10) 105 22. !

101

n!1(4 # 4n) "20,200

Find the first three terms of each arithmetic series described.

23. a1 ! 14, an ! #85, Sn ! #1207 24. a1 ! 1, an ! 19, Sn ! 100

14, 11, 8 1, 3, 5

25. n ! 16, an ! 15, Sn ! #120 26. n ! 15, an ! 5 , Sn ! 45

"30, "27, "24 , , 127. STACKING A health club rolls its towels and stacks them in layers on a shelf. Each

layer of towels has one less towel than the layer below it. If there are 20 towels on thebottom layer and one towel on the top layer, how many towels are stacked on the shelf?210 towels

28. BUSINESS A merchant places $1 in a jackpot on August 1, then draws the name of aregular customer. If the customer is present, he or she wins the $1 in the jackpot. If thecustomer is not present, the merchant adds $2 to the jackpot on August 2 and drawsanother name. Each day the merchant adds an amount equal to the day of the month. Ifthe first person to win the jackpot wins $496, on what day of the month was her or hisname drawn? August 31

3$

1$

4$5

2$3

Practice (Average)Arithmetic Series

NAME ______________________________________________ DATE ____________ PERIOD _____

11-211-2

Reading to Learn MathematicsArithmetic Series

NAME ______________________________________________ DATE ____________ PERIOD _____

11-211-2


Less

on

11-

2

Pre-Activity How do arithmetic series apply to amphitheaters?Read the introduction to Lesson 11-2 at the top of page 583 in your textbook.

Suppose that an amphitheater can seat 50 people in the first row and thateach row thereafter can seat 9 more people than the previous row. Usingthe vocabulary of arithmetic sequences, describe how you would find thenumber of people who could be seated in the first 10 rows. (Do not actuallycalculate the sum.) Sample answer: Find the first 10 terms of anarithmetic sequence with first term 50 and common difference9. Then add these 10 terms.

Reading the Lesson1. What is the relationship between an arithmetic sequence and the corresponding

arithmetic series? Sample answer: An arithmetic sequence is a list of termswith a common difference between successive terms. The correspondingarithmetic series is the sum of the terms of the sequence.

2. Consider the formula Sn ! (a1 " an). Explain the meaning of this formula in words.

Sample answer: To find the sum of the first n terms of an arithmeticsequence, find half the number of terms you are adding. Multiply thisnumber by the sum of the first term and the nth term.

3. a. What is the purpose of sigma notation?Sample answer: to write a series in a concise form

b. Consider the expression !12

i!2(4i # 2).

This form of writing a sum is called .

The variable i is called the .

The first value of i is .

The last value of i is .

How would you read this expression? The sum of 4i "2 as i goes from 2 to 12.

Helping You Remember4. A good way to remember something is to relate it to something you already know. How

can your knowledge of how to find the average of two numbers help you remember the formula Sn ! (a1 " an)? Sample answer: Rewrite the formula as

Sn ! n % . The average of the first and last terms is given by the

expression . The sum of the first n terms is the average of thefirst

a1 # an$2

a1 # an$2

n$2

122

index of summationsigma notation

n$2

Study Guide and InterventionGeometric Sequences

NAME ______________________________________________ DATE ____________ PERIOD _____

11-311-3


Less

on

11-

3

Geometric Sequences A geometric sequence is a sequence in which each term afterthe first is the product of the previous term and a constant called the constant ratio.

nth Term of a an ! a1 % r n # 1, where a1 is the first term, r is the common ratio, Geometric Sequence and n is any positive integer

Find the next twoterms of the geometric sequence 1200, 480, 192, .

Since ! 0.4 and ! 0.4, the

sequence has a common ratio of 0.4. Thenext two terms in the sequence are192(0.4) ! 76.8 and 76.8(0.4) ! 30.72.

192$480

480$1200

Write an equation for thenth term of the geometric sequence 3.6, 10.8, 32.4, .In this sequence a1 ! 3.6 and r ! 3. Use thenth term formula to write an equation.

an ! a1 % rn # 1 Formula for nth term

! 3.6 % 3n # 1 a1 ! 3.6, r ! 3

An equation for the nth term is an ! 3.6 % 3n # 1.


ExercisesExercises

Find the next two terms of each geometric sequence.

1. 6, 12, 24, 2. 180, 60, 20, 3. 2000, #1000, 500,

48, 96 , "250, 1254. 0.8, #2.4, 7.2, 5. 80, 60, 45, 6. 3, 16.5, 90.75,

"21.6, 64.8 33.75, 25.3125 499.125, 2745.1875Find the first five terms of each geometric sequence described.

7. a1 ! , r ! 3 8. a1 ! 240, r ! # 9. a1 ! 10, r !

, , 1, 3, 9 240, "180, 135, 10, 25, 62 , 156 ,

"101 , 75 390Find the indicated term of each geometric sequence.

10. a1 ! #10, r ! 4, n ! 2 11. a1 ! #6, r ! # , n ! 8 12. a3 ! 9, r ! #3, n ! 7

"40 729

13. a4 ! 16, r ! 2, n ! 10 14. a4 ! #54, r ! #3, n ! 6 15. a1 ! 8, r ! , n ! 5

1024 "486Write an equation for the nth term of each geometric sequence.

16. 500, 350, 245, 17. 8, 32, 128, 18. 11, #24.2, 53.24, 500 % 0.7n"1 8 % 4n" 1 11 % ("2.2)n " 1

128$

2$3

3$

1$2

5$

15$

1$

1$

1$

1$

1$

5$2

3$4

1$9

20$

20$


Find the next two terms of each geometric sequence.

1. #15, #30, #60, "120, "240 2. 80, 40, 20, 10, 5

3. 90, 30, 10, , 4. #1458, 486, #162, 54, "18

5. , , , , 6. 216, 144, 96, 64,

Find the first five terms of each geometric sequence described.

7. a1 ! #1, r ! #3 8. a1 ! 7, r ! #4

"1, 3, "9, 27, "81 7, "28, 112, "448, 17929. a1 ! # , r ! 2 10. a1 ! 12, r !

" , " , " , " , " 12, 8, , ,

Find the indicated term of each geometric sequence.

11. a1 ! 5, r ! 3, n ! 6 1215 12. a1 ! 20, r ! #3, n ! 6 "4860

13. a1 ! #4, r ! #2, n ! 10 2048 14. a8 for # , # , # , "

15. a12 for 96, 48, 24, 16. a1 ! 8, r ! , n ! 9

17. a1 ! #3125, r ! # , n ! 9 " 18. a1 ! 3, r ! , n ! 8

Write an equation for the nth term of each geometric sequence.

19. 1, 4, 16, an ! (4)n " 1 20. #1, #5, #25, an ! "1(5)n " 1

21. 1, , , an ! " #n " 1

22. #3, #6, #12, an ! "3(2)n " 1

23. 7, #14, 28, an ! 7("2)n " 1 24. #5, #30, #180, an ! "5(6)n " 1

Find the geometric means in each sequence.

25. 3, , , , 768 12, 48, 192 26. 5, , , , 1280 &20, 80, &32027. 144, , , , 9 28. 37,500, , , , , #12&72, 36, &18 "7500, 1500, "300, 60

29. BIOLOGY A culture initially contains 200 bacteria. If the number of bacteria doublesevery 2 hours, how many bacteria will be in the culture at the end of 12 hours? 12,800

30. LIGHT If each foot of water in a lake screens out 60% of the light above, what percent ofthe light passes through 5 feet of water? 1.024%

31. INVESTING Raul invests $1000 in a savings account that earns 5% interest compoundedannually. How much money will he have in the account at the end of 5 years? $1276.28

???????

??????

1$1$4

1$2

3$$1$10

1$125

1$5

1$1$2

3$

625$1$10

1$50

1$250

64$

32$

16$

16$

8$

4$

2$

1$

2$3

1$3

128$

81$

27$9$16

3$8

1$4

10$

10$

Practice (Average)Geometric Sequences

NAME ______________________________________________ DATE ____________ PERIOD _____

11-311-3

Jessica Rudolph

Reading to Learn MathematicsGeometric Sequences

NAME ______________________________________________ DATE ____________ PERIOD _____

11-311-3


Less

on

11-

3

Pre-Activity How do geometric sequences apply to a bouncing ball?Read the introduction to Lesson 11-3 at the top of page 588 in your textbook.

Suppose that you drop a ball from a height of 4 feet, and that each time itfalls, it bounces back to 74% of the height from which it fell. Describe howwould you find the height of the third bounce. (Do not actually calculate theheight of the bounce.)

Sample answer: Multiply 4 by 0.74 three times.

Reading the Lesson

1. Explain the difference between an arithmetic sequence and a geometric sequence.

Sample answer: In an arithmetic sequence, each term after the first isfound by adding the common difference to the previous term. In ageometric sequence, each term after the first is found by multiplying theprevious term by the common ratio.

2. Consider the formula an ! a1 % rn # 1.

a. What is this formula used to find? a particular term of a geometric sequenceb. What do each of the following represent?

an: the nth term a1: the first termr: the common ratio n: a positive integer that indicates which term you are finding

3. a. In the sequence 5, 8, 11, 14, 17, 20, the numbers 8, 11, 14, and 17 are

between 5 and 20.

b. In the sequence 12, 4, , , , the numbers 4, , and are

between 12 and .

Helping You Remember

4. Suppose that your classmate Ricardo has trouble remembering the formula an ! a1 % rn # 1

correctly. He thinks that the formula should be an ! a1 % rn. How would you explain to

him that he should use rn # 1 rather than rn in the formula?

Sample answer: Each term after the first in a geometric sequence isfound by multiplying the previous term by r. There are n " 1 termsbefore the nth term, so you would need to multiply by r a total of n " 1times, not n times, to get the nth term.

4$27

geometric means

4$9

4$3

4$27

4$9

4$3

arithmetic means

Study Guide and InterventionGeometric Series

NAME ______________________________________________ DATE ____________ PERIOD _____

11-411-4


Less

on

11-

4

Geometric Series A geometric series is the indicated sum of consecutive terms of ageometric sequence.

Sum of a The sum Sn of the first n terms of a geometric series is given byGeometric Series Sn ! or Sn ! , where r & 1.

a1 # a1rn$$1 # ra1(1 # r n)$$1 # r

Find the sum of the firstfour terms of the geometric sequence for which a1 ! 120 and r ! .

Sn ! Sum formula

S4 ! n ! 4, a1 ! 120, r ! $13$

" 177.78 Use a calculator.The sum of the series is 177.78.

120#1 # #$13$$4$

$$1 # $13$

a1(1 # rn)

$$1 # r

1$3

Find the sum of the

geometric series !7

j!14 % 3 j " 2.

Since the sum is a geometric series, you canuse the sum formula.

Sn ! Sum formula

S7 ! n ! 7, a1 ! $43$, r ! 3

" 1457.33 Use a calculator.The sum of the series is 1457.33.

$43$(1 # 37)$1 # 3

a1(1 # rn)

$$1 # r


ExercisesExercises

Find Sn for each geometric series described.

1. a1 ! 2, an ! 486, r ! 3 2. a1 ! 1200, an ! 75, r ! 3. a1 ! , an ! 125, r ! 5

728 2325 156.24

4. a1 ! 3, r ! , n ! 4 5. a1 ! 2, r ! 6, n ! 4 6. a1 ! 2, r ! 4, n ! 6

4.44 518 2730

7. a1 ! 100, r ! # , n ! 5 8. a3 ! 20, a6 ! 160, n! 8 9. a4 ! 16, a7 ! 1024, n ! 10

68.75 1275 87,381.25

Find the sum of each geometric series.

10. 6 " 18 " 54 " to 6 terms 11. " " 1 " to 10 terms

2184 255.75

12. !8

j!42 j 13. !

7

k!13 % 2k # 1

496 381

1$2

1$4

1$2

1$3

1$25

1$2


Specific Terms You can use one of the formulas for the sum of a geometric series to helpfind a particular term of the series.

Study Guide and Intervention (continued)Geometric Series

NAME ______________________________________________ DATE ____________ PERIOD _____

11-411-4

Find a1 in a geometricseries for which S6 ! 441 and r ! 2.

Sn ! Sum formula

441 ! S6 ! 441, r ! 2, n ! 6

441 ! Subtract.

a1 ! Divide.

a1 ! 7 Simplify.

The first term of the series is 7.

441$63

#63a1$#1

a1(1 # 26)$$1 # 2

a1(1 # rn)

$$1 # r

Find a1 in a geometricseries for which Sn ! 244, an ! 324, and r! "3.Since you do not know the value of n, use thealternate sum formula.

Sn ! Alternate sum formula

244 ! Sn ! 244, an ! 324, r ! #3

244 ! Simplify.

976 ! a1 " 972 Multiply each side by 4.a1 ! 4 Subtract 972 from each side.

The first term of the series is 4.

a1 " 972$$4

a1 # (324)(#3)$$1 # (#3)

a1 # anr$$1 # r


Example 3Example 3 Find a4 in a geometric series for which Sn ! 796.875, r ! , and n ! 8.First use the sum formula to find a1.

Sn ! Sum formula

796.875 ! S8 ! 796.875, r ! , n ! 8

796.875 ! Use a calculator.

a1 ! 400

Since a4 ! a1 % r3, a4 ! 400#$12$$

3! 50. The fourth term of the series is 50.

Find the indicated term for each geometric series described.

1. Sn ! 726, an ! 486, r ! 3; a1 6 2. Sn ! 850, an ! 1280, r ! #2; a1 "10

3. Sn ! 1023.75, an ! 512, r ! 2; a1 4. Sn ! 118.125, an ! #5.625, r ! # ; a1 180

5. Sn ! 183, r ! #3, n ! 5; a1 3 6. Sn ! 1705, r ! 4, n ! 5; a1 5

7. Sn ! 52,084, r ! #5, n ! 7; a1 4 8. Sn ! 43,690, r ! , n ! 8; a1 32, 768

9. Sn ! 381, r ! 2, n ! 7; a4 24

1$4

1$2

1$

0.99609375a1$$0.5

1$2

a1#1 # #$12$$8$

$$1 # $12$

a1(1 # rn)

$$1 # r

1$2

ExercisesExercises


Find Sn for each geometric series described.

1. a1 ! 2, a6 ! 64, r ! 2 126 2. a1 ! 160, a6 ! 5, r ! 315

3. a1 ! #3, an ! #192, r ! #2 "129 4. a1 ! #81, an ! #16, r ! # "55

5. a1 ! #3, an ! 3072, r ! #4 2457 6. a1 ! 54, a6 ! , r !

7. a1 ! 5, r ! 3, n ! 9 49,205 8. a1 ! #6, r ! #1, n ! 21 "6

9. a1 ! #6, r ! #3, n ! 7 "3282 10. a1 ! #9, r ! , n ! 4 "

11. a1 ! , r ! 3, n ! 10 12. a1 ! 16, r ! #1.5, n ! 6 "66.5

Find the sum of each geometric series.

13. 162 " 54 " 18 " to 6 terms 14. 2 " 4 " 8 " to 8 terms 510

15. 64 # 96 " 144 # to 7 terms 463 16. # " 1 # to 6 terms "

17. !8

n!1(#3)n # 1 "1640 18. !

9

n!15(#2)n # 1 855 19. !

5

n!1#1(4)n # 1 "341

20. !6

n!1# $n # 1 21. !

10

n!12560# $n # 1 5115 22. !

4

n!19# $n # 1

Find the indicated term for each geometric series described.

23. Sn ! 1023, an ! 768, r ! 4; a1 3 24. Sn ! 10,160, an ! 5120, r ! 2; a1 80

25. Sn ! #1365, n ! 12, r ! #2; a1 1 26. Sn ! 665, n ! 6, r ! 1.5; a1 32

27. CONSTRUCTION A pile driver drives a post 27 inches into the ground on its first hit.

Each additional hit drives the post the distance of the prior hit. Find the total distance

the post has been driven after 5 hits. 70 in.

28. COMMUNICATIONS Hugh Moore e-mails a joke to 5 friends on Sunday morning. Eachof these friends e-mails the joke to 5 of her or his friends on Monday morning, and so on.Assuming no duplication, how many people will have heard the joke by the end ofSaturday, not including Hugh? 97,655 people

1$

2$3

65$2$3

1$2

63$1$2

182$1$3

1$9

728$

29,524$1$3

65$2$3

728$1$3

2$9

2$3

1$2

Practice (Average)Geometric Series

NAME ______________________________________________ DATE ____________ PERIOD _____

11-411-4

Reading to Learn MathematicsGeometric Series

NAME ______________________________________________ DATE ____________ PERIOD _____

11-411-4


Less

on

11-

4

Pre-Activity How is e-mailing a joke like a geometric series?Read the introduction to Lesson 11-4 at the top of page 594 in your textbook.

Suppose that you e-mail the joke on Monday to five friends, rather thanthree, and that each of those friends e-mails it to five friends on Tuesday,and so on. Write a sum that shows that total number of people, includingyourself, who will have read the joke by Thursday. (Write out the sumusing plus signs rather than sigma notation. Do not actually find the sum.)1 # 5 # 25 # 125

Use exponents to rewrite the sum you found above. (Use an exponent ineach term, and use the same base for all terms.)50 # 51 # 52 # 53

Reading the Lesson

1. Consider the formula Sn ! .

a. What is this formula used to find? the sum of the first n terms of ageometric series

b. What do each of the following represent?

Sn: the sum of the first n termsa1: the first term r: the common ratio

c. Suppose that you want to use the formula to evaluate 3 # 1 " # " . Indicate

the values you would substitute into the formula in order to find Sn. (Do not actuallycalculate the sum.)

n ! a1 ! r ! rn !

d. Suppose that you want to use the formula to evaluate the sum !6

n!18(#2)n # 1. Indicate

the values you would substitute into the formula in order to find Sn. (Do not actuallycalculate the sum.)

n ! a1 ! r ! rn !

Helping You Remember

2. This lesson includes three formulas for the sum of the first n terms of a geometric series.All of these formulas have the same denominator and have the restriction r & 1. How canthis restriction help you to remember the denominator in the formulas?Sample answer: If r ! 1, then r " 1 ! 0. Because division by 0 isundefined, a formula with r " 1 in the denominator will not apply when r ! 1.

("2)6 or 64"286

""$13$#5 or "$2

143$"$

13$35

1$27

1$9

1$3

a1(1 # rn)

$$1 # r

Glencoe/McGraw-Hill A2 Glencoe Algebra 2

Answers (Lesson 11-1)

Stu

dy

Gu

ide

and I

nte

rven

tion

Arith

met

ic S

eque

nces

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD__

___

11-1

11-1

Gl

enco

e/M

cGra

w-Hi

ll63

1Gl

enco

e Al

gebr

a 2

Lesson 11-1

Ari

thm

etic

Seq

uenc

esA

n ar

ithm

etic

seq

uenc

e is

a s

eque

nce

of n

umbe

rs in

whi

ch e

ach

term

afte

r th

e fir

st t

erm

is fo

und

by a

ddin

g th

e co

mm

on d

iffe

renc

e to

the

pre

cedi

ng t

erm

.

nth T

erm

of a

n a n!

a 1"

(n#

1)d,

whe

re a

1is

the

first

term

, dis

the

com

mon

diffe

renc

e,

Arith

met

ic S

eque

nce

and

nis

any

posit

ive in

tege

r

Fin

d t

he

nex

t fo

ur

term

s of

th

e ar

ith

met

ic s

equ

ence

7,

11,1

5,

.F

ind

the

com

mon

dif

fere

nce

by s

ubtr

acti

ngtw

o co

nsec

utiv

e te

rms.

11 #

7 !

4 an

d 15

#11

!4,

so d!

4.

Now

add

4 t

o th

e th

ird

term

of t

he s

eque

nce,

and

then

con

tinu

e ad

ding

4 u

ntil

the

four

term

s ar

e fo

und.

The

nex

t fo

ur t

erm

s of

the

sequ

ence

are

19,

23,2

7,an

d 31

.

Fin

d t

he

thir

teen

th t

erm

of t

he

arit

hm

etic

seq

uen

ce w

ith

a1!

21an

d d!"

6.U

se t

he fo

rmul

a fo

r th

e nt

h te

rm o

f an

arit

hmet

ic s

eque

nce

wit

h a 1!

21,n!

13,

and

d!#

6.a n!

a 1"

(n#

1)d

Form

ula fo

r nth

term

a 13!

21 "

(13 #

1)(#

6)n!

13, a

1!

21, d!#

6a 1

3!#

51Si

mpli

fy.

The

thi

rtee

nth

term

is #

51.

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exam

ple

3Ex

ampl

e 3

Wri

te a

n e

quat

ion

for

th

e n

th t

erm

of

the

arit

hm

etic

seq

uen

ce

"14

,"5,

4,13

,.

In t

his

sequ

ence

a1!#

14 a

nd d!

9.U

se t

he f

orm

ula

for

a nto

wri

te a

n eq

uati

on.

a n!

a 1"

(n#

1)d

Form

ula fo

r the

nth

term

!#

14 "

(n#

1)9

a 1!#

14, d!

9!#

14 "

9n#

9Di

stribu

tive

Prop

erty

!9n#

23Si

mpli

fy.

Fin

d t

he

nex

t fo

ur

term

s of

eac

h a

rith

met

ic s

equ

ence

.

1.10

6,11

1,11

6,

2.#

28,#

31,#

34,

3.

207,

194,

181,

12

1,12

6,13

1,13

6"

37,"

40,"

43,"

4616

8,15

5,14

2,12

9

Fin

d t

he

firs

t fi

ve t

erm

s of

eac

h a

rith

met

ic s

equ

ence

des

crib

ed.

4.a 1!

101,

d!

95.

a 1!#

60,d!

46.

a 1!

210,

d!#

4010

1,11

0,11

9,12

8,13

7"

60,"

56,"

52,"

48,"

4421

0,17

0,13

0,90

,50

Fin

d t

he

ind

icat

ed t

erm

of

each

ari

thm

etic

seq

uen

ce.

7.a 1!

4,d!

6,n!

1482

8.a 1!#

4,d!#

2,n!

12"

269.

a 1!

80,d!#

8,n!

21"

8010

.a10

for

0,#

3,#

6,#

9,"

27

Wri

te a

n e

quat

ion

for

th

e n

th t

erm

of

each

ari

thm

etic

seq

uen

ce.

11.1

8,25

,32,

39,

12

.#11

0,#

85,#

60,#

35,

13

.6.2

,8.1

,10.

0,11

.9,

7n#

1125

n"

135

1.9n#

4.3

Exer

cises

Exer

cises

Gl

enco

e/M

cGra

w-Hi

ll63

2Gl

enco

e Al

gebr

a 2

Ari

thm

etic

Mea

ns

The

ari

thm

etic

mea

ns

of a

n ar

ithm

etic

seq

uenc

e ar

e th

e te

rms

betw

een

any

two

nons

ucce

ssiv

e te

rms

of t

he s

eque

nce.

To fi

nd t

he k

arit

hmet

ic m

eans

bet

wee

n tw

o te

rms

of a

seq

uenc

e,us

e th

e fo

llow

ing

step

s.

Step

1Le

t the

two

term

s giv

en b

e a 1

and

a n, w

here

n!

k"2.

Step

2Su

bstitu

te in

the

form

ula a

n!

a 1"

(n#

1)d.

Step

3So

lve fo

r d, a

nd u

se th

at v

alue

to fi

nd th

e ka

rithm

etic

mea

ns:

a 1"

d, a

1"

2d,

, a 1"

kd.

Fin

d t

he

five

ari

thm

etic

mea

ns

betw

een

37

and

121

.Yo

u ca

n us

e th

e nt

h te

rm f

orm

ula

to f

ind

the

com

mon

dif

fere

nce.

In t

he s

eque

nce,

37,

,,

,,

,121

,,a

1is

37

and

a 7is

121

.

a n!

a 1"

(n#

1)d

Form

ula fo

r the

nth

term

121 !

37 "

(7 #

1)d

a 1!

37, a

7!

121,

n!

712

1 !

37 "

6dSi

mpli

fy.84

!6d

Subt

ract

37 fr

om e

ach

side.

d!

14Di

vide

each

side

by

6.

Now

use

the

val

ue o

f dto

fin

d th

e fi

ve a

rith

met

ic m

eans

.37

!51

!65

!79

!93

!10

7 !

121

"14

"

14 "

14 "

14 "

14

"14

The

ari

thm

etic

mea

ns a

re 5

1,65

,79,

93,a

nd 1

07.

Fin

d t

he

arit

hm

etic

mea

ns

in e

ach

seq

uen

ce.

1.5,

,,

,#3

2.18

,,

,,#

23.

16,

,,3

73,

1,"

113

,8,3

23,3

04.

108,

,,

,,4

85.#

14,

,,

,#30

6.29

,,

,,8

996

,84,

72,6

0"

18,"

22,"

2644

,59,

747.

61,

,,

,,1

168.

45,

,,

,,

,81

72,8

3,94

,105

51,5

7,63

,69,

759.#

18,

,,

,14

10.#

40,

,,

,,

,#82

"10

,"2,

6"

47,"

54,"

61,"

68,"

7511

.100

,,

,235

12.8

0,,

,,

,#30

145,

190

58,3

6,14

,"8

13.4

50,

,,

,570

14.2

7,,

,,

,,5

748

0,51

0,54

032

,37,

42,4

7,52

15.1

25,

,,

,185

16.2

30,

,,

,,

,128

140,

155,

170

213,

196,

179,

162,

145

17.#

20,

,,

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7018

.48,

,,

,100

58,1

36,2

14,2

9261

,74,

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Stu

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(con

tinu

ed)

Arith

met

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eque

nces

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD__

___

11-1

11-1

Exam

ple

Exam

ple

Exer

cises

Exer

cises


An

swer

s


Skil

ls P

ract

ice

Arith

met

ic S

eque

nces

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD__

___

11-1

11-1

Gl

enco

e/M

cGra

w-Hi

ll63

3Gl

enco

e Al

gebr

a 2

Lesson 11-1

Fin

d t

he

nex

t fo

ur

term

s of

eac

h a

rith

met

ic s

equ

ence

.

1.7,

11,1

5,

19,2

3,27

,31

2.#

10,#

5,0,

5,

10,1

5,20

3.10

1,20

2,30

3,

404,

505,

606,

707

4.15

,7,#

1,"

9,"

17,"

25,"

33

5.#

67,#

60,#

53,

6.#

12,#

15,#

18,

"46

,"39

,"32

,"25

"21

,"24

,"27

,"30

Fin

d t

he

firs

t fi

ve t

erm

s of

eac

h a

rith

met

ic s

equ

ence

des

crib

ed.

7.a 1!

6,d!

96,

15,2

4,33

,42

8.a 1!

27,d!

427

,31,

35,3

9,43

9.a 1!#

12,d!

5"

12,"

7,"

2,3,

810

.a1!

93,d!#

1593

,78,

63,4

8,33

11.a

1!#

64,d!

1112

.a1!#

47,d!#

20"

64,"

53,"

42,"

31,"

20"

47,"

67,"

87,"

107,"

127

Fin

d t

he

ind

icat

ed t

erm

of

each

ari

thm

etic

seq

uen

ce.

13.a

1!

2,d!

6,n!

1268

14.a

1!

18,d!

2,n!

832

15.a

1!

23,d!

5,n!

2313

316

.a1!

15,d!#

1,n!

25"

9

17.a

31fo

r 34

,38,

42,

154

18.a

42fo

r 27

,30,

33,

150

Com

ple

te t

he

stat

emen

t fo

r ea

ch a

rith

met

ic s

equ

ence

.

19.5

5 is

the

th

ter

m o

f 4,

7,10

,.

1820

.163

is t

he

th t

erm

of #

5,2,

9,

.25

Wri

te a

n e

quat

ion

for

th

e n

th t

erm

of

each

ari

thm

etic

seq

uen

ce.

21.4

,7,1

0,13

,a n!

3n#

122

.#1,

1,3,

5,

a n!

2n"

3

23.#

1,3,

7,11

,a n!

4n"

524

.7,2

,#3,#

8,

a n!"

5n#

12

Fin

d t

he

arit

hm

etic

mea

ns

in e

ach

seq

uen

ce.

25.6

,,

,,3

814

,22,

3026

.63,

,,

,147

84,1

05,1

26?

??

??

?

??

Gl

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4Gl

enco

e Al

gebr

a 2

Fin

d t

he

nex

t fo

ur

term

s of

eac

h a

rith

met

ic s

equ

ence

.

1.5,

8,11

,14

,17,

20,2

32.#

4,#

6,#

8,"

10,"

12,"

14,"

163.

100,

93,8

6,

79,7

2,65

,58

4.#

24,#

19,#

14,

"9,"

4,1,

65.

,6,

,11,

,1

6,,2

16.

4.8,

4.1,

3.4,

2.

7,2,

1.3,

0.6

Fin

d t

he

firs

t fi

ve t

erm

s of

eac

h a

rith

met

ic s

equ

ence

des

crib

ed.

7.a 1!

7,d!

78.

a 1!#

8,d!

2

7,14

,21,

28,3

5"

8,"

6,"

4,"

2,0

9.a 1!#

12,d!#

410

.a1!

,d!

"12

,"16

,"20

,"24

,"28

,1,

,2,

11.a

1!#

,d!#

12.a

1!

10.2

,d!#

5.8

","

,","

,"10

.2,4

.4,"

1.4,"

7.2,"

13

Fin

d t

he

ind

icat

ed t

erm

of

each

ari

thm

etic

seq

uen

ce.

13.a

1!

5,d!

3,n!

1032

14.a

1!

9,d!

3,n!

2993

15.a

18fo

r #

6,#

7,#

8,

."

2316

.a37

for

124,

119,

114,

."

5617

.a1!

,d!#

,n!

10"

18.a

1!

14.2

5,d!

0.15

,n!

3118

.75

Com

ple

te t

he

stat

emen

t fo

r ea

ch a

rith

met

ic s

equ

ence

.

19.1

66 is

the

th

ter

m o

f 30

,34,

38,

3520

.2 is

the

th

ter

m o

f ,

,1,

8

Wri

te a

n e

quat

ion

for

th

e n

th t

erm

of

each

ari

thm

etic

seq

uen

ce.

21.#

5,#

3,#

1,1,

a n!

2n"

722

.#8,#

11,#

14,#

17,

a n!"

3n"

523

.1,#

1,#

3,#

5,

a n!"

2n#

324

.#5,

3,11

,19,

a n!

8n"

13

Fin

d t

he

arit

hm

etic

mea

ns

in e

ach

seq

uen

ce.

25.#

5,,

,,1

1"

1,3,

726

.82,

,,

,18

66,5

0,34

27.E

DU

CA

TIO

NT

revo

r K

oba

has

open

ed a

n E

nglis

h L

angu

age

Scho

ol in

Ise

hara

,Jap

an.

He

bega

n w

ith

26 s

tude

nts.

If h

e en

rolls

3 n

ew s

tude

nts

each

wee

k,in

how

man

y w

eeks

will

he

have

101

stu

dent

s?26

wk

28.S

ALA

RIE

SYo

land

a in

terv

iew

ed f

or a

job

that

pro

mis

ed h

er a

sta

rtin

g sa

lary

of

$32,

000

wit

h a

$125

0 ra

ise

at t

he e

nd o

f ea

ch y

ear.

Wha

t w

ill h

er s

alar

y be

dur

ing

her

sixt

h ye

arif

she

acc

epts

the

job?

$38,

250

??

??

??

4 $ 53 $ 5

??

18 $ 53 $ 5

9 $ 5

13 $ 611 $ 6

3 $ 27 $ 6

5 $ 6

1 $ 35 $ 6

5 $ 23 $ 2

1 $ 2

1 $ 21 $ 2

37 $ 227 $ 2

17 $ 27 $ 2

Pra

ctic

e (A

vera

ge)

Arith

met

ic S

eque

nces

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD__

___

11-1

11-1



Rea

din

g t

o L

earn

Math

emati

csAr

ithm

etic

Seq

uenc

es

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD__

___

11-1

11-1

Gl

enco

e/M

cGra

w-Hi

ll63

5Gl

enco

e Al

gebr

a 2

Lesson 11-1

Pre-

Act

ivit

yH

ow a

re a

rith

met

ic s

equ

ence

s re

late

d t

o ro

ofin

g?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 11

-1 a

t th

e to

p of

pag

e 57

8 in

you

r te

xtbo

ok.

Des

crib

e ho

w y

ou w

ould

find

the

num

ber

of s

hing

les

need

ed fo

r th

e fif

teen

thro

w.(

Do

not

actu

ally

cal

cula

te t

his

num

ber.)

Exp

lain

why

you

r m

etho

d w

illgi

ve t

he c

orre

ct a

nsw

er.

Sam

ple

answ

er:A

dd 3

tim

es 1

4 to

2.T

his

work

s be

caus

e th

e fir

st ro

w ha

s 2

shin

gles

and

3 m

ore

are

adde

d 14

tim

es to

go

from

the

first

row

to th

e fif

teen

th ro

w.

Rea

din

g t

he

Less

on

1.C

onsi

der

the

form

ula

a n!

a 1"

(n#

1)d.

a.W

hat

is t

his

form

ula

used

to

find

?a

parti

cula

r ter

m o

f an

arith

met

ic s

eque

nce

b.W

hat

do e

ach

of t

he f

ollo

win

g re

pres

ent?

a n:

the

nth

term

a 1:

the

first

term

n:

a po

sitiv

e in

tege

r tha

t ind

icat

es w

hich

term

you

are

find

ing

d:th

e co

mm

on d

iffer

ence

2.C

onsi

der

the

equa

tion

an!#

3n"

5.

a.W

hat

does

thi

s eq

uati

on r

epre

sent

?Sa

mpl

e an

swer

:It g

ives

the

nth

term

of

an a

rithm

etic

seq

uenc

e wi

th fi

rst t

erm

2 a

nd c

omm

on d

iffer

ence

"3.

b.Is

the

gra

ph o

f th

is e

quat

ion

a st

raig

ht li

ne?

Exp

lain

you

r an

swer

.Sa

mpl

ean

swer

:No;

the

grap

h is

a s

et o

f poi

nts

that

fall

on a

line

,but

the

poin

ts d

o no

t fill

the

line.

c.T

he f

unct

ions

rep

rese

nted

by

the

equa

tion

s a n!#

3n"

5 an

d f(

x) !#

3x"

5 ar

eal

ike

in t

hat

they

hav

e th

e sa

me

form

ula.

How

are

the

y di

ffer

ent?

Sam

ple

answ

er:T

hey

have

diff

eren

t dom

ains

.The

dom

ain

of th

e fir

st fu

nctio

nis

the

set o

f pos

itive

inte

gers

.The

dom

ain

of th

e se

cond

func

tion

isth

e se

t of a

ll re

al n

umbe

rs.

Hel

pin

g Y

ou

Rem

emb

er3.

A g

ood

way

to

rem

embe

r so

met

hing

is t

o ex

plai

n it

to

som

eone

els

e.Su

ppos

e th

at y

our

clas

smat

e Sh

ala

has

trou

ble

rem

embe

ring

the

for

mul

a a n!

a 1"

(n#

1)d

corr

ectl

y.Sh

eth

inks

tha

t th

e fo

rmul

a sh

ould

be

a n!

a 1"

nd.H

ow w

ould

you

exp

lain

to

her

that

she

shou

ld u

se (n#

1)d

rath

er t

han

ndin

the

for

mul

a?Sa

mpl

e an

swer

:Eac

h te

rmaf

ter t

he fi

rst i

n an

arit

hmet

ic s

eque

nce

is fo

und

by a

ddin

g d

to th

epr

evio

us te

rm.Y

ou w

ould

add

don

ce to

get

to th

e se

cond

term

,twi

ce to

get t

o th

e th

ird te

rm,a

nd s

o on

.So

dis

add

ed n"

1 tim

es,n

ot n

times

,to

get

the

nth

term

.

Gl

enco

e/M

cGra

w-Hi

ll63

6Gl

enco

e Al

gebr

a 2

Fibo

nacc

i Seq

uenc

eL

eona

rdo

Fib

onac

ci f

irst

dis

cove

red

the

sequ

ence

of

num

bers

nam

ed f

or h

imw

hile

stu

dyin

g ra

bbit

s.H

e w

ante

d to

kno

w h

ow m

any

pair

s of

rab

bits

wou

ldbe

pro

duce

d in

nm

onth

s,st

arti

ng w

ith

a si

ngle

pai

r of

new

born

rab

bits

.He

mad

e th

e fo

llow

ing

assu

mpt

ions

.

1.N

ewbo

rn r

abbi

ts b

ecom

e ad

ults

in o

ne m

onth

.

2.E

ach

pair

of

rabb

its

prod

uces

one

pai

r ea

ch m

onth

.

3.N

o ra

bbit

s di

e.

Let

Fn

repr

esen

t th

e nu

mbe

r of

pai

rs o

f ra

bbit

s at

the

end

of n

mon

ths.

If y

oube

gin

wit

h on

e pa

ir o

f ne

wbo

rn r

abbi

ts,F

0!

F1!

1.T

his

pair

of

rabb

its

wou

ld p

rodu

ce o

ne p

air

at t

he e

nd o

f th

e se

cond

mon

th,s

o F

2!

1 "

1,or

2.

At

the

end

of t

he t

hird

mon

th,t

he f

irst

pai

r of

rab

bits

wou

ld p

rodu

ce a

noth

erpa

ir.T

hus,

F3!

2 "

1,or

3.

The

cha

rt b

elow

sho

ws

the

num

ber

of r

abbi

ts e

ach

mon

th f

or s

ever

al m

onth

s.

Sol

ve.

1.St

arti

ng w

ith

a si

ngle

pai

r of

new

born

rab

bits

,how

man

y pa

irs

of r

abbi

tsw

ould

the

re b

e at

the

end

of

12 m

onth

s?

233

2.W

rite

the

fir

st 1

0 te

rms

of t

he s

eque

nce

for

whi

ch F

0!

3,F

1!

4,an

d F n!

F n#

2"

F n#

1.

3,4,

7,11

,18,

29,4

7,76

,123

,199

,322

3.W

rite

the

fir

st 1

0 te

rms

of t

he s

eque

nce

for

whi

ch F

0!

1,F

1!

5,F n

!F n

#2"

F n"

1.

1,5,

6,11

,17,

28,4

5,73

,118

,191

,309

Mon

thAd

ult P

airs

Newb

orn

Pairs

Tota

lF 0

01

1F 1

10

1F 2

11

2F 3

21

3F 4

32

5F 5

53

8

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD__

___

11-1

11-1


An

swer

s


Stu

dy

Gu

ide

and I

nte

rven

tion

Arith

met

ic S

erie

s

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD__

___

11-2

11-2

Gl

enco

e/M

cGra

w-Hi

ll63

7Gl

enco

e Al

gebr

a 2

Lesson 11-2

Ari

thm

etic

Ser

ies

An

arit

hm

etic

ser

ies

is t

he s

um o

f co

nsec

utiv

e te

rms

of a

nar

ithm

etic

seq

uenc

e.

Sum

of a

n Th

e su

m S

nof

the

first

nte

rms

of a

n ar

ithm

etic

serie

s is

given

by

the

form

ulaAr

ithm

etic

Ser

ies

S n!

$n 2$ [2a

1"

(n#

1)d]

or S

n!

$n 2$(a

1"

a n)

Fin

d S

nfo

r th

ear

ith

met

ic s

erie

s w

ith

a1!

14,

an!

101,

and

n!

30.

Use

the

sum

for

mul

a fo

r an

ari

thm

etic

seri

es.

Sn!

(a1"

a n)

Sum

form

ula

S30!

(14 "

101)

n!

30, a

1!

14, a

n!

101

!15

(115

)Si

mpli

fy.!

1725

Mult

iply.

The

sum

of

the

seri

es is

172

5.

30 $ 2n $ 2

Fin

d t

he

sum

of

all

pos

itiv

e od

d i

nte

gers

les

s th

an 1

80.

The

ser

ies

is 1

"3 "

5 "

"

179.

Fin

d n

usin

g th

e fo

rmul

a fo

r th

e nt

h te

rm o

fan

ari

thm

etic

seq

uenc

e.

a n!

a 1"

(n#

1)d

Form

ula fo

r nth

term

179 !

1 "

(n#

1)2

a n!

179,

a1!

1, d!

217

9 !

2n#

1Si

mpli

fy.18

0 !

2nAd

d 1

to e

ach

side.

n!

90Di

vide

each

side

by

2.

The

n us

e th

e su

m f

orm

ula

for

an a

rith

met

icse

ries

.

Sn!

(a1"

a n)

Sum

form

ula

S90!

(1 "

179)

n!

90, a

1!

1, a

n!

179

!45

(180

)Si

mpli

fy.!

8100

Mult

iply.

The

sum

of

all p

osit

ive

odd

inte

gers

less

than

180

is 8

100.

90 $ 2n $ 2

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Fin

d S

nfo

r ea

ch a

rith

met

ic s

erie

s d

escr

ibed

.

1.a 1!

12,a

n!

100,

2.a 1!

50,a

n!#

50,

3.a 1!

60,a

n!#

136,

n!

12

672

n!

150

n!

50"

1900

4.a 1!

20,d!

4,5.

a 1!

180,

d!#

8,6.

a 1!#

8,d!#

7,a n!

112

1584

a n!

6818

60a n!#

71"

395

7.a 1!

42,n!

8,d!

68.

a 1!

4,n!

20,d!

29.

a 1!

32,n!

27,d!

3

504

555

1917

Fin

d t

he

sum

of

each

ari

thm

etic

ser

ies.

10.8

"6 "

4 "

"#

10"

1011

.16 "

22 "

28 "

"

112

1088

12.#

45 "

(#41

) "

(#37

) "

"

35"

105

Fin

d t

he

firs

t th

ree

term

s of

eac

h a

rith

met

ic s

erie

s d

escr

ibed

.

13.a

1!

12,a

n!

174,

14.a

1!

80,a

n!#

115,

15.a

1!

6.2,

a n!

12.6

,S

n!

1767

12,2

1,30

Sn!#

245

80,6

5,50

Sn!

84.6

6.2,

7.0,

7.8

1 $ 2

Gl

enco

e/M

cGra

w-Hi

ll63

8Gl

enco

e Al

gebr

a 2

Sig

ma

No

tati

on

A s

hort

hand

not

atio

n fo

r re

pres

enti

ng a

ser

ies

mak

es u

se o

f th

e G

reek

lett

er

.The

sig

ma

not

atio

nfo

r th

e se

ries

6 "

12 "

18 "

24 "

30 is

!5 n!16

n.

Eva

luat

e !18 k!

1(3k#

4).

The

sum

is a

n ar

ithm

etic

ser

ies

wit

h co

mm

on d

iffe

renc

e 3.

Subs

titu

ting

k!

1 an

d k!

18in

to t

he e

xpre

ssio

n 3k"

4 gi

ves

a 1!

3(1)

"4 !

7 an

d a 1

8!

3(18

) "

4 !

58.T

here

are

18

ter

ms

in t

he s

erie

s,so

n!

18.U

se t

he f

orm

ula

for

the

sum

of

an a

rith

met

ic s

erie

s.

Sn!

(a1"

a n)

Sum

form

ula

S18!

(7 "

58)

n!

18, a

1!

7, a

n!

58

!9(

65)

Sim

plify.

!58

5M

ultipl

y.

So !18 k!

1(3

k"

4) !

585.

Fin

d t

he

sum

of

each

ari

thm

etic

ser

ies.

1.!20 n!

1(2

n"

1)

2.!25 n!

5(x#

1)

3.!18 k!

1(2

k#

7)

440

294

216

4.!75 r!1

0(2

r#

200)

5.

!15 x!1(6

x"

3)

6.!50 t!1

(500

#6t

)

"75

9076

517

,350

7.!80 k!

1(1

00 #

k)8.

!85n!

20(n#

100)

9.

!200 s!13s

4760

"31

3560

,300

10.!28

m!

14(2

m#

50)

11.!36 p!

1(5

p#

20)

12.!32 j!1

2(2

5 #

2j)

"12

026

10"

399

13.!42

n!

18(4

n#

9)

14.!50

n!

20(3

n"

4)

15.!44 j!5

(7j#

3)

2775

3379

6740

18 $ 2n $ 2

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Arith

met

ic S

erie

s

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD__

___

11-2

11-2

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Arith

met

ic S

erie

s

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD__

___

11-2

11-2

Gl

enco

e/M

cGra

w-Hi

ll63

9Gl

enco

e Al

gebr

a 2

Lesson 11-2

Fin

d S

nfo

r ea

ch a

rith

met

ic s

erie

s d

escr

ibed

.

1.a 1!

1,a n!

19,n!

1010

02.

a 1!#

5,a n!

13,n!

728

3.a 1!

12,a

n!#

23,n!

8"

444.

a 1!

7,n!

11,a

n!

6740

7

5.a 1!

5,n!

10,a

n!

3218

56.

a 1!#

4,n!

10,a

n!#

22"

130

7.a 1!#

8,d!#

5,n!

12"

426

8.a 1!

1,d!

3,n!

1533

0

9.a 1!

100,

d!#

7,a n!

3768

510

.a1!#

9,d!

4,a n!

2790

11.d!

2,n!

26,a

n!

4244

212

.d!#

12,n!

11,a

n!#

5288

Fin

d t

he

sum

of

each

ari

thm

etic

ser

ies.

13.1

"4 "

7 "

10 "

"

4333

014

.5 "

8 "

11 "

14 "

"

3218

5

15.3

"5 "

7 "

9 "

"

1999

16.#

2 "

(#5)

"(#

8) "

"

(#20

)"

77

17. !5 n!

1(2

n#

3)15

18. !1

8

n!

1(1

0 "

3n)

693

19. !1

0

n!

2(4

n"

1)22

520

. !12

n!

5(4

#3n

)"

172

Fin

d t

he

firs

t th

ree

term

s of

eac

h a

rith

met

ic s

erie

s d

escr

ibed

.

21.a

1!

4,a n!

31,S

n!

175

4,7,

1022

.a1!#

3,a n!

41,S

n!

228"

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NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD__

___

11-2

11-2


An

swer

s


Rea

din

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o L

earn

Math

emati

csAr

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Ser

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NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD__

___

11-2

11-2

Gl

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1Gl

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Lesson 11-2

Pre-

Act

ivit

yH

ow d

o ar

ith

met

ic s

erie

s ap

ply

to

amp

hit

hea

ters

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 11

-2 a

t th

e to

p of

pag

e 58

3 in

you

r te

xtbo

ok.

Supp

ose

that

an

amph

ithe

ater

can

sea

t 50

peo

ple

in t

he f

irst

row

and

tha

tea

ch r

ow t

here

afte

r ca

n se

at 9

mor

e pe

ople

tha

n th

e pr

evio

us r

ow.U

sing

the

voca

bula

ry o

f ar

ithm

etic

seq

uenc

es,d

escr

ibe

how

you

wou

ld f

ind

the

num

ber

of p

eopl

e w

ho c

ould

be

seat

ed in

the

fir

st 1

0 ro

ws.

(Do

not

actu

ally

calc

ulat

e th

e su

m.)

Sam

ple

answ

er:F

ind

the

first

10

term

s of

an

arith

met

ic s

eque

nce

with

firs

t ter

m 5

0 an

d co

mm

on d

iffer

ence

9.Th

en a

dd th

ese

10 te

rms.

Rea

din

g t

he

Less

on

1.W

hat

is t

he r

elat

ions

hip

betw

een

an a

rith

met

ic s

eque

nce

and

the

corr

espo

ndin

gar

ithm

etic

ser

ies?

Sam

ple

answ

er:A

n ar

ithm

etic

seq

uenc

e is

a li

st o

f ter

ms

with

a c

omm

on d

iffer

ence

bet

ween

suc

cess

ive

term

s.Th

e co

rresp

ondi

ngar

ithm

etic

ser

ies

is th

e su

m o

f the

term

s of

the

sequ

ence

.2.

Con

side

r th

e fo

rmul

a S

n!

(a1"

a n).

Exp

lain

the

mea

ning

of

this

for

mul

a in

wor

ds.

Sam

ple

answ

er:T

o fin

d th

e su

m o

f the

firs

t nte

rms

of a

n ar

ithm

etic

sequ

ence

,fin

d ha

lf th

e nu

mbe

r of t

erm

s yo

u ar

e ad

ding

.Mul

tiply

this

num

ber b

y th

e su

m o

f the

firs

t ter

m a

nd th

e nt

h te

rm.

3.a.

Wha

t is

the

pur

pose

of

sigm

a no

tati

on?

Sam

ple

answ

er:t

o wr

ite a

ser

ies

in a

con

cise

form

b.C

onsi

der

the

expr

essi

on !12 i!

2(4

i#

2).

Thi

s fo

rm o

f w

riti

ng a

sum

is c

alle

d .

The

var

iabl

e i

is c

alle

d th

e .

The

fir

st v

alue

of i

is

.

The

last

val

ue o

f iis

.

How

wou

ld y

ou r

ead

this

exp

ress

ion?

The

sum

of 4

i"2

as i

goes

from

2 to

12.

Hel

pin

g Y

ou

Rem

emb

er4.

A g

ood

way

to

rem

embe

r so

met

hing

is t

o re

late

it t

o so

met

hing

you

alr

eady

kno

w.H

owca

n yo

ur k

now

ledg

e of

how

to

find

the

ave

rage

of

two

num

bers

hel

p yo

u re

mem

ber

the

form

ula

Sn!

(a1"

a n)?

Sam

ple

answ

er:R

ewrit

e th

e fo

rmul

a as

S n!

n%

.The

ave

rage

of t

he fi

rst a

nd la

st te

rms

is g

iven

by

the

expr

essi

on

.The

sum

of t

he fi

rst n

term

s is

the

aver

age

of th

e fir

st

and

last

term

s m

ultip

lied

by th

e nu

mbe

r of t

erm

s.

a 1#

a n$

2

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2n $ 2

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For

th

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his

pag

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u w

ill

nee

d t

o u

se t

he

Pyt

hag

orea

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heo

rem

an

d t

he

form

ula

s fo

r th

e ar

ea o

f a

tria

ngl

e an

d a

tra

pez

oid

.

En

rich

men

t

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD__

___

11-2

11-2

1.A

rec

tang

le m

easu

res

5 by

12

unit

s.T

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left

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ut o

ff a

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own

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e di

agra

m.

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ind

the

area

A(x

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sha

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x) !

60 "

(5 "

x)(6

"x)

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ind

xan

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A(x

) is

am

axim

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hat

happ

ens

to t

he

cut-

off

tria

ngle

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d 2x!

10;t

he tr

iang

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ll no

t exi

st.

3.T

he c

oord

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es o

f th

e ve

rtic

es o

f a

tria

ngle

are

A(0

,0),

B(1

1,0)

,and

C

(0,1

1).A

line

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he t

rian

gle

into

tw

o re

gion

s ha

ving

equ

al a

rea.

a.W

hat

are

the

coor

dina

tes

of p

oint

D?

(k,1

1 "

k)b.

Wri

te a

nd s

olve

an

equa

tion

for

find

ing

the

valu

e of

k.

$1 2$ k(1

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k) !

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2.A

tri

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side

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leng

ths

a,a,

and

bis

isos

cele

s.T

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s ar

e cu

t of

f so

that

the

rem

aini

ng p

enta

gon

has

five

equa

l sid

es o

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ngth

x.T

he v

alue

of x

can

be f

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his

equa

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b#

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a!

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x &

4.46

b.C

an a

be e

qual

to

2b?

Yes,

but i

t wou

ld n

ot b

epo

ssib

le to

hav

e a

pent

agon

of

the

type

des

crib

ed.

4.In

side

a s

quar

e ar

e fi

ve c

ircl

es w

ith

the

sam

e ra

dius

.

a.C

onne

ct t

he c

ente

r of

the

top

left

cir

cle

to t

he c

ente

r of

the

bot

tom

rig

ht c

ircl

e.E

xpre

ss t

his

leng

th in

ter

ms

of r

.4r

b.D

raw

the

squ

are

wit

h ve

rtic

es a

t th

ece

nter

s of

the

fou

r ou

tsid

e ci

rcle

s.E

xpre

ss t

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iago

nal o

f th

is s

quar

ein

ter

ms

of r

and

a.

(a"

2r)$

2%

ra

bxxx

x

xa

x

y

AB

C

D

x ! k

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12

5

x



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nte

rven

tion

Geom

etric

Seq

uenc

es

NAM

E__

____

____

____

____

____

____

____

____

____

____

____

DATE

____

____

____

PERI

OD__

___

11-3

11-3

Gl

enco

e/M

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w-Hi

ll64

3Gl

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a 2

Lesson 11-3

Geo

met

ric

Seq

uen

ces

A g

eom

etri

c se

quen

ceis

a s

eque

nce

in w

hich

eac

h te

rm a

fter

the

firs

t is

the

pro

duct

of

the

prev

ious

ter

m a

nd a

con

stan

t ca

lled

the

con

stan

t ra

tio.

nth T

erm

of a

a n!

a 1%

rn#

1 , wh

ere

a 1is

the

first

sequences series

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