Download - Sequences Series
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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
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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
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 5/15 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
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
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
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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
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Glencoe/McGraw-Hill 634 Glencoe Algebra 2
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
??????
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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
Glencoe/McGraw-Hill 635 Glencoe Algebra 2
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.
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Study Guide and InterventionArithmetic Series
NAME ______________________________________________ DATE ____________ PERIOD _____
11-211-2
Glencoe/McGraw-Hill 637 Glencoe Algebra 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.
Sn ! (a1 " an) Sum formula
S90 ! (1 " 179) n ! 90, a1 ! 1, an ! 179
! 45(180) Simplify.! 8100 Multiply.
The sum of all positive odd integers lessthan 180 is 8100.
90$2
n$2
Example 1Example 1 Example 2Example 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
-
Glencoe/McGraw-Hill 638 Glencoe Algebra 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.
Sn ! (a1 " an) Sum formula
S18 ! (7 " 58) n ! 18, a1 ! 7, an ! 58
! 9(65) Simplify.! 585 Multiply.
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
-
Glencoe/McGraw-Hill 640 Glencoe Algebra 2
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
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2$3
Practice (Average)Arithmetic Series
NAME ______________________________________________ DATE ____________ PERIOD _____
11-211-2
-
Reading to Learn MathematicsArithmetic Series
NAME ______________________________________________ DATE ____________ PERIOD _____
11-211-2
Glencoe/McGraw-Hill 641 Glencoe Algebra 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
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Study Guide and InterventionGeometric Sequences
NAME ______________________________________________ DATE ____________ PERIOD _____
11-311-3
Glencoe/McGraw-Hill 643 Glencoe Algebra 2
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$1200
Write an equation for thenth term of the geometric sequence 3.6, 10.8, 32.4, .In this sequence a1 ! 3.6 and r ! 3. Use thenth term formula to write an equation.
an ! a1 % rn # 1 Formula for nth term
! 3.6 % 3n # 1 a1 ! 3.6, r ! 3
An equation for the nth term is an ! 3.6 % 3n # 1.
Example 1Example 1 Example 2Example 2
ExercisesExercises
Find the next two terms of each geometric sequence.
1. 6, 12, 24, 2. 180, 60, 20, 3. 2000, #1000, 500,
48, 96 , "250, 1254. 0.8, #2.4, 7.2, 5. 80, 60, 45, 6. 3, 16.5, 90.75,
"21.6, 64.8 33.75, 25.3125 499.125, 2745.1875Find the first five terms of each geometric sequence described.
7. a1 ! , r ! 3 8. a1 ! 240, r ! # 9. a1 ! 10, r !
, , 1, 3, 9 240, "180, 135, 10, 25, 62 , 156 ,
"101 , 75 390Find the indicated term of each geometric sequence.
10. a1 ! #10, r ! 4, n ! 2 11. a1 ! #6, r ! # , n ! 8 12. a3 ! 9, r ! #3, n ! 7
"40 729
13. a4 ! 16, r ! 2, n ! 10 14. a4 ! #54, r ! #3, n ! 6 15. a1 ! 8, r ! , n ! 5
1024 "486Write an equation for the nth term of each geometric sequence.
16. 500, 350, 245, 17. 8, 32, 128, 18. 11, #24.2, 53.24, 500 % 0.7n"1 8 % 4n" 1 11 % ("2.2)n " 1
128$
2$3
3$
1$2
5$
15$
1$
1$
1$
1$
1$
5$2
3$4
1$9
20$
20$
-
Glencoe/McGraw-Hill 646 Glencoe Algebra 2
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
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Practice (Average)Geometric Sequences
NAME ______________________________________________ DATE ____________ PERIOD _____
11-311-3
Jessica Rudolph
-
Reading to Learn MathematicsGeometric Sequences
NAME ______________________________________________ DATE ____________ PERIOD _____
11-311-3
Glencoe/McGraw-Hill 647 Glencoe Algebra 2
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
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Study Guide and InterventionGeometric Series
NAME ______________________________________________ DATE ____________ PERIOD _____
11-411-4
Glencoe/McGraw-Hill 649 Glencoe Algebra 2
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
Example 1Example 1 Example 2Example 2
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
-
Glencoe/McGraw-Hill 650 Glencoe Algebra 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$$1 # r
Example 1Example 1 Example 2Example 2
Example 3Example 3 Find a4 in a geometric series for which Sn ! 796.875, r ! , and n ! 8.First use the sum formula to find a1.
Sn ! Sum formula
796.875 ! S8 ! 796.875, r ! , n ! 8
796.875 ! Use a calculator.
a1 ! 400
Since a4 ! a1 % r3, a4 ! 400#$12$$
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
-
Glencoe/McGraw-Hill 652 Glencoe Algebra 2
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
Glencoe/McGraw-Hill 653 Glencoe Algebra 2
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
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PERI
OD__
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11-1
11-1
Gl
enco
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w-Hi
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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,
,,
,,3
7018
.48,
,,
,100
58,1
36,2
14,2
9261
,74,
87?
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Stu
dy
Gu
ide
and I
nte
rven
tion
(con
tinu
ed)
Arith
met
ic S
eque
nces
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD__
___
11-1
11-1
Exam
ple
Exam
ple
Exer
cises
Exer
cises
-
Glencoe/McGraw-Hill A3 Glencoe Algebra 2
An
swer
s
Answers (Lesson 11-1)
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
enco
e/M
cGra
w-Hi
ll63
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
-
Glencoe/McGraw-Hill A4 Glencoe Algebra 2
Answers (Lesson 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
-
Glencoe/McGraw-Hill A5 Glencoe Algebra 2
An
swer
s
Answers (Lesson 11-2)
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
-
Glencoe/McGraw-Hill A6 Glencoe Algebra 2
Answers (Lesson 11-2)
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"
3,1,
5
23.n!
10,a
n!
41,S
n!
230
5,9,
1324
.n!
19,a
n!
85,S
n!
760"
5,0,
5
Gl
enco
e/M
cGra
w-Hi
ll64
0Gl
enco
e Al
gebr
a 2
Fin
d S
nfo
r ea
ch a
rith
met
ic s
erie
s d
escr
ibed
.
1.a 1!
16,a
n!
98,n!
1374
12.
a 1!
3,a n!
36,n!
1223
4
3.a 1!#
5,a n!#
26,n!
8"
124
4.a 1!
5,n!
10,a
n!#
13"
40
5.a 1!
6,n!
15,a
n!#
22"
120
6.a 1!#
20,n!
25,a
n!
148
1600
7.a 1!
13,d!#
6,n!
21"
987
8.a 1!
5,d!
4,n!
1127
5
9.a 1!
5,d!
2,a n!
3328
510
.a1!#
121,
d!
3,a n!
5"
2494
11.d!
0.4,
n!
10,a
n!
3.8
2012
.d!#
,n!
16,a
n!
4478
4
Fin
d t
he
sum
of
each
ari
thm
etic
ser
ies.
13.5
"7 "
9 "
11 "
"
2719
214
.#4 "
1 "
6 "
11 "
"
9187
015
.13 "
20 "
27 "
"
272
5415
16.8
9 "
86 "
83 "
80 "
"
2013
08
17. !4 n!
1(1
#2n
)"
1618
. !6 j!1(5
"3n
)93
19. !5 n!
1(9
#4n
)"
15
20. !1
0
k!
4(2
k"
1)10
521
. !8 n!3(5
n#
10)
105
22. !10
1
n!
1(4
#4n
)"
20,2
00
Fin
d t
he
firs
t th
ree
term
s of
eac
h a
rith
met
ic s
erie
s d
escr
ibed
.
23.a
1!
14,a
n!#
85,S
n!#
1207
24.a
1!
1,a n!
19,S
n!
100
14,1
1,8
1,3,
5
25.n!
16,a
n!
15,S
n!#
120
26.n!
15,a
n!
5,S
n!
45
"30
,"27
,"24
,,1
27.S
TAC
KIN
GA
hea
lth
club
rol
ls it
s to
wel
s an
d st
acks
the
m in
laye
rs o
n a
shel
f.E
ach
laye
r of
tow
els
has
one
less
tow
el t
han
the
laye
r be
low
it.I
f th
ere
are
20 t
owel
s on
the
bott
om la
yer
and
one
tow
el o
n th
e to
p la
yer,
how
man
y to
wel
s ar
e st
acke
d on
the
she
lf?
210
towe
ls28
.BU
SIN
ESS
A m
erch
ant
plac
es $
1 in
a ja
ckpo
t on
Aug
ust
1,th
en d
raw
s th
e na
me
of a
regu
lar
cust
omer
.If
the
cust
omer
is p
rese
nt,h
e or
she
win
s th
e $1
in t
he ja
ckpo
t.If
the
cust
omer
is n
ot p
rese
nt,t
he m
erch
ant
adds
$2
to t
he ja
ckpo
t on
Aug
ust
2 an
d dr
aws
anot
her
nam
e.E
ach
day
the
mer
chan
t ad
ds a
n am
ount
equ
al t
o th
e da
y of
the
mon
th.I
fth
e fi
rst
pers
on t
o w
in t
he ja
ckpo
t w
ins
$496
,on
wha
t da
y of
the
mon
th w
as h
er o
r hi
sna
me
draw
n?Au
gust
31
3 $ 51 $ 5
4 $ 5
2 $ 3
Pra
ctic
e (A
vera
ge)
Arith
met
ic S
erie
s
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD__
___
11-2
11-2
-
Glencoe/McGraw-Hill A7 Glencoe Algebra 2
An
swer
s
Answers (Lesson 11-2)
Rea
din
g t
o L
earn
Math
emati
csAr
ithm
etic
Ser
ies
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD__
___
11-2
11-2
Gl
enco
e/M
cGra
w-Hi
ll64
1Gl
enco
e Al
gebr
a 2
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
a 1#
a n$
2n $ 2
122in
dex
of s
umm
atio
nsi
gma
nota
tion
n $ 2
Gl
enco
e/M
cGra
w-Hi
ll64
2Gl
enco
e Al
gebr
a 2
Geom
etric
Puz
zlers
For
th
e p
robl
ems
on t
his
pag
e,yo
u w
ill
nee
d t
o u
se t
he
Pyt
hag
orea
nT
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
heup
per
left
cor
ner
is c
ut o
ff a
s sh
own
inth
e di
agra
m.
a.F
ind
the
area
A(x
) of
the
sha
ded
pent
agon
.A(
x) !
60 "
(5 "
x)(6
"x)
b.F
ind
xan
d 2x
so t
hat
A(x
) is
am
axim
um.W
hat
happ
ens
to t
he
cut-
off
tria
ngle
?x!
5 an
d 2x!
10;t
he tr
iang
lewi
ll no
t exi
st.
3.T
he c
oord
inat
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
x!
kcu
ts t
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
1 #
11 "
k) !
22;
k!
11 "
$77%
2.A
tri
angl
e w
ith
side
s of
leng
ths
a,a,
and
bis
isos
cele
s.T
wo
tria
ngle
s ar
e cu
t of
f so
that
the
rem
aini
ng p
enta
gon
has
five
equa
l sid
es o
f le
ngth
x.T
he v
alue
of x
can
be f
ound
usi
ng t
his
equa
tion
.(2
b#
a)x2"
(4a2#
b2)(
2x#
a) !
0
a.F
ind
xw
hen
a!
10 a
nd b!
12.
x &
4.46
b.C
an a
be e
qual
to
2b?
Yes,
but i
t wou
ld n
ot b
epo
ssib
le to
hav
e a
pent
agon
of
the
type
des
crib
ed.
4.In
side
a s
quar
e ar
e fi
ve c
ircl
es w
ith
the
sam
e ra
dius
.
a.C
onne
ct t
he c
ente
r of
the
top
left
cir
cle
to t
he c
ente
r of
the
bot
tom
rig
ht c
ircl
e.E
xpre
ss t
his
leng
th in
ter
ms
of r
.4r
b.D
raw
the
squ
are
wit
h ve
rtic
es a
t th
ece
nter
s of
the
fou
r ou
tsid
e ci
rcle
s.E
xpre
ss t
he d
iago
nal o
f th
is s
quar
ein
ter
ms
of r
and
a.
(a"
2r)$
2%
ra
bxxx
x
xa
x
y
AB
C
D
x ! k
2x
12
5
x
-
Glencoe/McGraw-Hill A8 Glencoe Algebra 2
Answers (Lesson 11-3)
Stu
dy
Gu
ide
and I
nte
rven
tion
Geom
etric
Seq
uenc
es
NAM
E__
____
____
____
____
____
____
____
____
____
____
____
DATE
____
____
____
PERI
OD__
___
11-3
11-3
Gl
enco
e/M
cGra
w-Hi
ll64
3Gl
enco
e Al
gebr
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