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UNIT ONE

SEQUENCES AND SERIES

MATH 421A

15 HOURS

Revised Dec 7, 00

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UNIT 1: Sequences and series

Previous Knowledge

With the implementation of APEF Mathematics at the Intermediate level, students will have shouldbe able to:

- Grade 7 - describe patterns in tables, graphs and equations

- Grade 7 - represent patterns using a mathematical equation

- Grade 8 - demonstrate an understanding of the Pythagorean Theorem using models

- Grade 8 - apply the Pythagorean Theorem in problems

- Grade 9 - describe patterns in tables, graphs and equations

Overview:

- an overview of various types of sequences

- arithmetic sequences and series

- geometric sequences

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SCO: By the end of grade10 students will beexpected to:

A6 apply properties of numbers when operating and expressing equations

Elaborations - Instructional Strategies/SuggestionsSolving Linear Equations (4.6)

Students should be invited to do a simple review on solving simplelinear equations such as:

Ex:

3 6 4x x− = +

3 2 7 12 4( )− = + −z z z

2 2 1 6 2 3 3 1 3( ) ( ) ( )s s s+ + − + − = −

The perimeter of the rectangle below is 14 units. Find the length of thesides.

25

12

110

y− =

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Worthwhile Tasks for Instruction and/or Assessment Suggested Resources

Solving Linear Equations (4.6)Pencil/PaperThe perimeter of the rectangle below is 14 units. Find thelength of the sides.

Pencil/PaperSolve and check:a) 3 6 4x x− = +

b) 3 2 7 12 4( )− = + −z z z

c) 2 2 1 6 2 3 3 1 3( ) ( ) ( )s s s+ + − + − = −

d) 25

12

110

y− =

Group ActivityThe length of a rectangle is 21 cm. The length is 1 cm morethan twice the width. What is the width of the rectangle?

DiscussionWrite a word problem that can be solved using the followingequations:a) 0 25 6. x =

b) x x+ + =( )3 22

Solving Linear Equations

Math Power 10 p.179 #9-37 oddApplications p.180 #41,43,45-47

p.185 #9-37 oddApplications p.186 #71,73,75

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SCO: By the end of grade10 students will beexpected to:

A20 represent arithmetic and geometric sequences as ordered pairs and discrete graphs

B35 develop, analyze and apply algorithms to determine terms in a sequence

Elaborations - Instructional Strategies/SuggestionsNote to Teachers: The TI-83 graphing may be too advanced for the startof the year. These examples are for the teachers’ information. Teachersmay choose to do a little of the TI-83 work with students if they sodesire.Introduction (2.1) As you look around, patterns are noticeable everywhere. From natureexamples such as pine cones, chambered nautilus shells(math powerp.2), leaf construction and sunflowers. There are also many man-madeexamples in the real world. Patterns and predictability are what thehuman eye finds aesthetically pleasing.Invite students to discuss various number patterns some of which theyhave met previously in Junior High. Examples that should come up inthe discussion are:@ arithmetic having a common difference where the graph is linear tn = a + (n!1)dexample: 2, 6, 10, 14, 18,...@ geometric with no common difference but a common ratio tn = arn!1

example: !3, 6, !12, 24,...- neither where the above patterns are not in evidenceexample: 1, 5, 10, 17,...An example of a simple arithmetic sequence might be the followingpictorial pattern composed of:

On the TI-83 the graph could be done by pressingStat 1:Edit if necessary clear L1 and L2 by : cursor up to the top of L1 then pressclear enter Repeat if necessary for L2 now you can enter the data in the table and get

Worthwhile Tasks for Instruction and/or Assessment Suggested Resources

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Introduction (2.1)Pencil/Paper/CommunicationWhich type of sequence is each of the following; predict thenext two terms in each sequence.

1) Number of coloured squares in each term below

2) Perimeters of the squares in each term below

3)

4) Number of squares in each term below

IntroductionMathpower 10 p.52 #1 a,b,c

Problem Solving StrategiesMath Power 10 p.63 # 1-3 Green

Mathpower 10 p.65 #1-6 p.67 #41-44,46-48,50

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SCO: By the end of grade10 students will beexpected to:

B35 develop, analyze and apply algorithms to determine terms in a sequence

Elaborations - Instructional Strategies/SuggestionsIntroduction(cont’d) (2.1)

To graph this data; 2nd Stat Plot 1:Plot 1 and have the settings as shown

press window and adjust the settings as shown

Now press graph See p. 74 in Math Power for a second example.

Consider the sequence tn = n2. A useful technique for seeing patterns insequences is to look for a common difference between the terms. If thereis none, look for a common second difference and so on.

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Worthwhile Tasks for Instruction and/or Assessment Suggested Resources

Introduction(cont’d) (2.1)ManipulativesUse Link-a-cubes to create an arithmetic and geometricsequence. Explain your sequence to other members of yourgroup.

Ex:

Arithmetic - # of white squaresGeometric - total # of squares

ResearchDo an internet search on sequences in the real world. Perhapsyou can just look around at things in nature or things in yourhome, etc. What types of sequences did you find?

Pencil/PaperIdentify each of the following sequences as arithmetic,geometric or neither.1) 5, 2, !1, !4, ... (Arithmetic)

2) 2, 1, 1/2, 1/4, ... (Geometric)

3) 2, !6, 18, !54, ... (Geometric)

4) 1, 4, 9, 16, ... (Neither)

5) !2, 1, 6, 13, ... (Neither)

CommunicationIn your groups, come to a consensus as to why each of theabove sequences is what you have stated?

JournalWrite a brief description explaining the key features of eachtype of sequence: arithmetic and geometric.

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SCO: By the end of grade10 students will be expectedto:

A20 represent arithmetic and geometric sequences as ordered pairs and discrete graphs

B35 develop, analyze and apply algorithms to determine terms in a sequence

Elaborations - Instructional Strategies/SuggestionsIntroduction (cont’d) (2.1)An interesting example of a sequence that is neither arithmetic norgeometric is tn = n3.

An example of a geometric sequence is ;

Ageometric (exponential) sequence is a fixed number (base) raised to apower made of consecutive counting numbers or tn = arn!1.

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Worthwhile Tasks for Instruction and/or Assessment Suggested Resources

Introduction(cont’d) (2.1)

Pencil/Paper/TechnologyGenerate the first six terms of the sequence tn = 3A 2n!1 . Draw the graph.

TI-83 solution (steps on page 27-28)

Examining the table above we see that a = 3 and r =2 Using these in tn = arn!1 yields tn = 2 A 3n!1 .

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SCO: By the end of grade10 students will beexpected to:

B34 demonstrate how recursive rules relate to arithmetic and geometric sequences

B35 develop, analyze and apply algorithms to determine terms in a sequence

Elaborations - Instructional Strategies/Suggestions

Logic (2.2)Another NCTM initiative is to improve logical thinking skills. Onemethod to assist in this process is to organize information in tables.

Example: Look at a section of a tile floor that is 8 × 8. How manysquares are contained within this 8 × 8 section of the floor.

Manipulatives/Pencil/PaperOne method is to try to solve a simpler problem to discover a pattern.

Students can possibly see the pattern and get the answer of 204.

Arithmetic Sequences (2.5)Challenge students to explore arithmetic sequences in more detail.Goals for students to achieve are:

C find the formula for the general term tn = a + (n ! 1)d

C find the value of a specific term using the general term

C find the number of terms in a sequence

C find a specific number of arithmetic means between two terms in a sequence

C graph tn vs n

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Worthwhile Tasks for Instruction and/or Assessment Suggested Resources

Logic (2.2)Group workInvite students to work on some of the interesting problems inthe Suggested Resources

Arithmetic Sequences (2.5)Pencil/PaperComplete the following table. Let the first term be 3 and thecommon difference be 2.

Can you see the relationship between the number of the term and the term itself?

Solution tn = 2n + 1

Pencil/PaperComplete the following table. Let the first term be denoted as“a” and the common difference as “d”. Can you see therelationship between the number of the term and the termitself?

solution a + (n ! 1)d

See p.72 in Math Power 10

Problem Solving StrategiesMath Power 10 p.60 ex. p.61 #1,5

Arithmetic Sequences

Mathpower 10 p.75#23-29 odd p.75#33-37 odd Applications p.75#39-41,44,45,49 50,52,53,63,65

Math 10 p.8#5,9.10 p.14#4,8,11

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Worthwhile Tasks for Instruction and/or Assessment Suggested Resources

Arithmetic Sequences (cont’d) (2.5)Pencil/PaperHow many terms are in the sequence !5, 2, 9, ..., 184 ?Solution (28)

Pencil/PaperInsert four arithmetic means between 2 and 17.Solution (5,8,11,14)

JournalWrite a few sentences that would explain the term“arithmetic mean” to another person.

Pencil/Paper/TechnologyCreate an arithmetic sequence containing 5 terms, then graph tn vs n using graph paper or the Stat Plot feature on theTI-83.

Pencil/Paper/CommunicationSue invests $10,000 in an 8% Canada Savings Bond wherethe interest is paid yearly. Sue will earn simple interest on thistype of bond and at the end of the bond’s term it will be worthits initial face value. Complete the table below.

What kind of sequence has been generated in the table? Whatis the expression for the general term tn ?

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Worthwhile Tasks for Instruction and/or Assessment Suggested Resources

Arithmetic sequences (cont’d) (2.5)Pencil Paper/TechnologyIf an arithmetic sequence has a first term of 2 and a commondifference of 5, find the formula for the general term tn for thisspecific sequence.

TI-83 solutionBy substituting into the formula tn = a + (n ! 1)d we gettn = 2 + (n ! 1)5 =5n-3. Try to use the TI-83 to generate twolists: L1 for the number of the term and L2 for the term itself.If we want our list to include the first 20 terms then:To fill in L1:Stat 1:Edit Then clear the listswith the cursor on L1 at the very top of the L1 column2nd List < OPS 5:Seq then enter (x, x, 1, 20, 1) enter(set of all x, formula, from x= 1, to x=20, in steps of 1)

enter

To fill in L2:have the cursor at the top of L2 and enter 5L1 ! 3

Yielding

So if we wanted to see a specific term, say the 19th , we couldcursor down and see it on the screen

Problem Solving StrategiesMath Power 10 p.70 # 1,3,5,7,10,11

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SCO: By the end of grade10 students will be expectedto:

B36 develop, analyze and apply algorithms to determine the sum of a series

Elaborations - Instructional Strategies/Suggestions

Arithmetic Series (2.7)

Challenge student groups to discuss and propose solutions to #6p,79 in Mathpower 10. This will serve as a lead-in to arithmetic series. Allow time for students to read and discuss p.80-82 in Mathpower10 or p.17-19 in Math 10. Have the discussion focus on the logic ofthe two sum formulae. The logic used by Gauss when he was around 8years old is shown on p.81.

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Worthwhile Tasks for Instruction and/or Assessment Suggested Resources

Arithmetic Series (2.7)Pencil/Paper/TechnologyFind the sum of the first 20 terms of the arithmetic series:3 + 7 + 11 + ...

TI-83 solutionIn L1 enter the numbers from 1 to 20 : have cursor at top of L12nd List < OPS 5:Seq ( x,x,1,20,1) yields

next generate the sequence in L2have the cursor at the top of L2 and enter 3 +( L1 ! 1) 4

enter yields

now you can generate the sum in L3

2nd List 6: CumSum(L2)

if we press enter we will see

To see the sum of the first 20 terms cursor down and see

JournalDescribe a step-by-step procedure to calculate the sum of anarithmetic series using the TI-83.

Arithmetic Series

Mathpower 10 p.82-83 #4,10,15,16, 18,22,23,26, 27,29,31

Math 10 p.20 #4,7,8

Problem Solving StrategiesMath Power 10 p.79 # 2,4,6

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SCO: By the end of grade10 students will be expectedto:

A20 represent arithmetic and geometric sequences as ordered pairs and discrete graphs

B34 demonstrate how recursive rules relate to arithmetic and geometric sequences

B35 develop, analyze and apply algorithms to determine terms in a sequence

Elaborations - Instructional Strategies/Suggestions

Geometric Sequences (p.84)Allow student groups to discuss their understanding ofgeometric(exponential ) sequences. Record the results of each groupon the board.

C no common difference, first, second, third, etc.

C a common ratio

Note to teachers: This will be kept on a very introductory level. Noformulae will be developed in this section. This topic will be picked upagain in Grade 12.Example:You are re-negotiating your allowance with your parents. Theyweren’t born yesterday but luckily for you they never did sequenceswhen they were in school. (They probably had to walk 6 km each dayto school, up-hill both ways). You propose a deal to them: “give me 1cent the first day and double that every day.” How much will you geton the 30th day and also for the whole month?

TI-83 solution

to generate the allowance in L2 cursor to the top of L2 and enter

Press enter

to generate the cumulative sum cursor to the top of L32nd List < OPS 6: cumSum(L2)

To see the answers, cursor down

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Worthwhile Tasks for Instruction and/or Assessment Suggested Resources

Geometric Sequences (p.84)Pencil/Paper/TechnologyThe number of people on Earth doubles every 30 years. Thereare currently 6 billion people. How many will there be in 180years. Complete the following table and graph the results.Write an expression to represent the number of people after nyears.Solution

Geometric Sequences

Mathpower 10 p.84-85 #1,3 green

Math 10 p.27 #4,7,9,12

Problem Solving StrategiesMath Power 10 p.93 # 1,6

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Which type of sequence is each of the following; predict the next two terms in each sequence.

1) Number of coloured squares in each term below

2) Perimeters of the squares in each term below

3)

4) Number of squares in each term below