english language arts 10 – 2 web viewthere are 41 home games in the regular season. ......

MATHEMATICS 20-1

Sequences and Series

High School collaborative venture withEdmonton Christian, Harry Ainlay, J. Percy Page, Jasper Place, Millwoods Christian, Ross Sheppard and W. P. Wagner, M. E LaZerte, McNally, Queen Elizabeth, Strathcona and Victoria

Edm Christian High: Aaron TrimbleHarry Ainlay: Ben LuchkowHarry Ainlay: Darwin HoltHarry Ainlay: Lareina RezewskiHarry Ainlay: Mike ShrimptonJ. Percy Page: Debbie YoungerJasper Place: Matt KatesJasper Place: Sue DvorackMillwoods Christian: Patrick YpmaRoss Sheppard: Patricia ElderRoss Sheppard: Dean WallsW. P. Wagner: Amber SteinhauerM. E. LaZerte: Teena WoudstraQueen Elizabeth: David UnderwoodStrathcona: Christian Digout Victoria: Steven Dyck McNally: Neil Peterson

Facilitator: John Scammell (Consulting Services)Editor: Jim Reed (Contracted)

2010 - 2011

Mathematics 20-1 Sequences and Series of 36

TABLE OF CONTENTS

STAGE 1 DESIRED RESULTS PAGE

Big Idea

Enduring Understandings

Essential Questions

4

4

5

Knowledge

Skills

6

7

STAGE 2 ASSESSMENT EVIDENCE

Transfer TaskArena Plan

Teacher Notes for Transfer Task and RubricTransfer TaskRubricPossible Solution

8101214

STAGE 3 LEARNING PLANS

Lesson #1 Introduction to Patterns 16

Lesson #2 Arithmetic Series 20

Lesson #3 Geometric Sequences 23

Lesson #4 Geometric Series 26

Lesson #5 Infinite Geometric Series 30


Implementation note:Post the BIG IDEA in a prominentplace in your classroom and refer to it often.

Mathematics 20-1 Sequences and Series

STAGE 1 Desired Results

Big Idea:

The world is full of patterns to be discovered. Students will be able to recognize a pattern and continue modeling the sequence to make predictions for future elements.

Enduring Understandings:

Students will understand … Different types of sequences and series exist. We can use mathematics to model the pattern of the sequence or series.

By the end of the unit students should: Use concrete strategies to determine the pattern. Have an idea of where to start in breaking down the sequence/series. Recognize and apply patterns to familiar and unfamiliar situations (predictions). Know that a pattern exists. See patterns in life, application of patterns beyond geometric/arithmetic

sequences and series. Make predictions based on an observed pattern. Determine the pattern and identify relevant elements of geometric/arithmetic

sequences and series. Investigate or discover patterns and extend them.


Essential Questions:

Is anything in the universe truly random? Is chaos a pattern? What is the underlying structure in the pattern that allows sequences and series

to be expressed mathematically, concretely, symbolically, pictorially, and verbally in different terms depending on the context?

Can we recognize that there is universality to patterns that manifest themselves in different contexts in nature?

Additional questions to consider

Is there an underlying structure/connection that helps us identify that different patterns exist?

How are exceptions to the pattern a pattern itself? You have a routine M-F, but your routine is different on Saturday. The

exception on Saturday is a pattern in itself When we are looking for patterns, where do we start looking? Can we identify that we have an unconscious sense of patterns (rule of thirds)?

Can we learn to recognize and name these patterns? Can we recognize intrinsic/automatic patterns and acknowledge that they are in

fact patterns?


http://en.wikipedia.org/wiki/Rule_of_thirds

Knowledge:

EnduringUnderstanding

SpecificOutcomes

Description ofKnowledge

Students will understand…

Different types of sequences and series exist.

*RF9 RF10

Students will know …

the difference between arithmetic and geometric

the notation of sequences and series (a, d, n, r, tn)

the components required to finding the general term

the difference between convergent and divergent geometric series and what leads to convergence


We can use mathematics to model the pattern of the sequence or series.

RF9 RF10



the notation of sequences and series (a, d, n, r, tn)



8888I*RF = Relations and Functions


Implementation note:Teachers need to continually askthemselves, if their students are acquiring the knowledge and skills needed for the unit.

Skills:

EnduringUnderstanding

SpecificOutcomes

Description ofSkills


Different types of sequences and series exist.

* RF9 RF10

Students will be able to…

identify arithmetic and geometric sequences create a model for a problem/scenario calculate any specified parameter for a

sequence or series (a, d, n, r, tn, Sn) find the sum of a sequence or the individual

terms or a series



RF9 RF10

Students will be able to…



terms or a series calculate the infinite sum of a convergent

series

* RF = Relations and Functions


Implementation note:Teachers need to consider what performances and products will reveal evidence of understanding?What other evidence will be collected to reflectthe desired results?

STAGE 2 Assessment Evidence

1 Desired Results Desired Results

Arena Plan

Teacher Notes

This task is open-ended. Many different student responses are possible.

Students could research construction costs, arena designs, and ticketing practices online before beginning.

Have students read the whole thing before beginning. Part 3 relies on the answer to Part 2, and students should consider Part 3 as they work on Part 2.

Part 3 is difficult to do just with formulas because it mixes both arithmetic and geometric sequences. As a result, students may want to use a spreadsheet to answer part 3.

Students should be encouraged to present their proposals to Mr. Dogs in whatever format they like. It could be a verbal presentation, done on a poster, video or PowerPoint. They should consider the best way to present the proposal to Mr. Dogs.


Glossary

arithmetic sequence - A sequence for which the difference between successive terms is constant

arithmetic series - The sum of the terms of an arithmetic sequence

common difference - A constant that is added to each term to produce an arithmetic sequence

common ratio - A constant that is multiplied to each term to produce a geometric sequence

convergent sequence – A sequence in which the difference between two consecutive terms is equal to zero when n is large

convergent series – A series in which the sum is finite

divergent series – A series that is not convergent

first term – The first value in a/an arithmetic/geometric sequence/series.

general term - A function that describes all terms in a sequence

geometric sequence - A sequence in which the ratio of successive terms is constant

geometric series - The sum of the terms of a geometric sequence

infinite sequence - A sequence that does not end or have a final term

infinite geometric series – A geometric series that does not end or have a final term. An infinite geometric series may be convergent or divergent.

sequence - A set or list of numbers arranged in a definite order. A sequence is a function whose domain is a subset of the natural numbers, N, and whose range is a subset of the real numbers, R. The sequence itself shows the range of the function.


Arena Plan - Student Assessment Task

Scenario

Mr. Dogs has asked your company to design a seating plan for a new NHL arena. Currently his team plays in a rink like the one below.

Sample Arena

You must create a proposal to Mr. Dogs that outlines the following information. Support your proposal with appropriate mathematics.

Arena Plan - Student Assessment Task

1. Mr. Dogs wants the number of seats in the arena to be between 18 000 and 22 500. One ring of seats all the way around the rink is considered a row, and row 1 is considered to be the row closest to the ice. He wants the number of seats in each row to form an arithmetic sequence, increasing by the same number in each subsequent row. Your task is to decide on the total number of seats in the arena by designing a seating arrangement that has a reasonable number of rows by determining:

a. The number of seats in the first row.b. The number of rows required.c. The number of seats by which each row increases.d. The number of seats in the last row.e. The total number of seats in the arena.

2. In his current arena, Mr. Dogs charges $6000 per season for seats in rows 1-10, $4000 for season seats in rows 11-20, $3000 for season seats in rows 21-30, and $2000 for season seats in rows 31-40. He thinks that a more fair way to decide on season ticket prices is to use a geometric sequence, and decrease the price in each subsequent row by the same factor based on the price of the row in front of it. For your proposal

a. Determine a reasonable price per game for each seat in the first row.b. Determine the factor by which the cost of each seat per game will

decrease in each subsequent row from row 1.c. Determine the price per game of each seat in the last row.

3. There are 41 home games in the regular season. Given that he needs to sell every seat in the arena and generate at least $50 000 000 in revenue, determine the following:

a. The total revenue he will generate by selling all the seats in his rink at the prices you set above. You may have to adjust the prices you set above in order to generate at least $50 000 000 in revenue.

Your proposal can take any form, but must be supported by mathematics.

Glossary





convergent sequence – A sequence in which the difference between two consecutive terms is equal to zero when n is large

convergent series – A series in which the sum is finite

divergent series – A series that is not convergent








Assessment

Mathematics 20-1

Sequences and Series

Rubric

Level Excellent Proficient Adequate Limited Insufficient

Criteria 4 3 2 1 BlankMath ContentPart 1

All required elements are present and correct

All required elements are present but may contain minor errors

Some required elements are missing, or contain major errors

Most required elements are missing or incorrect

No score is awarded as there is no evidence given

Math ContentPart 2






Math Content Part 3






Presents Data

Presentation of data is clear, precise and accurate

Presentation of data is complete and unambiguous

Presentation of data is simplistic and plausible

Presentation of data is vague and inaccurate

Presentation of data is incomprehensible

Explains Choices

Provides insightful explanations

Provides logical explanations

Provides explanations that are complete but vague

Provides explanations that are incomplete or confusing.

No explanation is provided

When work is judged to be limited or insufficient, the teacher makes decisions.


Possible Solution to Arena Plan

A solution such as this could be presented in many different ways:

Number of Seats

We propose to have 460 seats in row 1, and increase the number of seats by 4 in each subsequent row. If we have 40 rows, the total number of seats in the arena will be 21 520, as shown below.

=

40 [2 ( 460 )+(40−1 ) 4 ]2

Ticket Price

We propose that the ticket price per game for seats in row 1 should be $400. Each subsequent row should receive an 8% decrease in this price, making the ticket price per game in row 40 a very reasonable $15.48.

Total Revenue

Based on our proposed model, Mr. Dogs can expect a total revenue of $98 868 825.80. A spreadsheet is useful in determining the total revenue based on the above model.


Row Seats $/GmRow's Revenue/GM Games

Row's Revenue/Season

1 460 $400.00 $184,000.00 41 $7,544,000.002 464 $368.00 $170,752.00 41 $7,000,832.003 468 $338.56 $158,446.08 41 $6,496,289.284 472 $311.48 $147,016.29 41 $6,027,668.075 476 $286.56 $136,401.22 41 $5,592,450.006 480 $263.63 $126,543.65 41 $5,188,289.757 484 $242.54 $117,390.33 41 $4,813,003.468 488 $223.14 $108,891.66 41 $4,464,557.929 492 $205.29 $101,001.47 41 $4,141,060.44

10 496 $188.86 $93,676.81 41 $3,840,749.3911 500 $173.76 $86,877.69 41 $3,561,985.3212 504 $159.85 $80,566.90 41 $3,303,242.7113 508 $147.07 $74,709.81 41 $3,063,102.2114 512 $135.30 $69,274.23 41 $2,840,243.4315 516 $124.48 $64,230.20 41 $2,633,438.2116 520 $114.52 $59,549.86 41 $2,441,544.2617 524 $105.36 $55,207.30 41 $2,263,499.3418 528 $96.93 $51,178.43 41 $2,098,315.7319 532 $89.17 $47,440.86 41 $1,945,075.0920 536 $82.04 $43,973.75 41 $1,802,923.7421 540 $75.48 $40,757.76 41 $1,671,068.1222 544 $69.44 $37,774.89 41 $1,548,770.6923 548 $63.88 $35,008.44 41 $1,435,346.0224 552 $58.77 $32,442.86 41 $1,330,157.1525 556 $54.07 $30,063.71 41 $1,232,612.3026 560 $49.75 $27,857.60 41 $1,142,161.6127 564 $45.77 $25,812.06 41 $1,058,294.3128 568 $42.10 $23,915.51 41 $980,535.9529 572 $38.74 $22,157.22 41 $908,445.8430 576 $35.64 $20,527.19 41 $841,614.7231 580 $32.79 $19,016.16 41 $779,662.5332 584 $30.16 $17,615.52 41 $722,236.3533 588 $27.75 $16,317.28 41 $669,008.5234 592 $25.53 $15,114.02 41 $619,674.8335 596 $23.49 $13,998.85 41 $573,952.8836 600 $21.61 $12,965.38 41 $531,580.5237 604 $19.88 $12,007.67 41 $492,314.4438 608 $18.29 $11,120.21 41 $455,928.8139 612 $16.83 $10,297.90 41 $422,214.0840 616 $15.48 $9,535.99 41 $390,975.75


Implementation note:

Each lesson is a conceptual unit and is not intended to be taught on a one lesson per block basis. Each represents a concept to be covered and can take anywhere from part of a class to several classes to complete.

STAGE 3 Learning Plans

Lesson 1

Introduction to Patterns

STAGE 1

BIG IDEA: The world is full of patterns to be discovered. Students will be able to recognize a pattern and continue modeling the sequence to make predictions for future elements.

ENDURING UNDERSTANDINGS:

Students will understand …


ESSENTIAL QUESTIONS:

Is anything in the universe truly random? Is chaos a pattern?

What is the underlying structure in the pattern that allows sequences and series to be expressed mathematically, concretely, symbolically, pictorially, and verbally in different terms depending on the context?


KNOWLEDGE:



SKILLS:

Students will be able to …


sequence or series (a, d, n, r, tn, Sn)


Lesson Summary

Use patterns to generate the general term of an arithmetic sequence and examine each parameter.

Lesson Plan

ThinkWhy do we need/where do we see sequences and series?

Explore PatternsGive examples of arithmetic, geometric and other sequences. Ask students to find the next two terms and determine a rule for the pattern.

HookWhat animal are you (your birth year - http://www.chinese.new-year.co.uk/calendar.htm)? What does your animal tell you about your personality traits, and is it accurate?What about other family members?

Lesson

Have students explore and discover the general term by doing the following:

o Have students choose an animal in the Chinese calendar.o Students will list the first 6 terms for the years of the animal they choose.o Students will create a formula for their sequence (not necessarily using the

parameters from the general term).o Begin to lead the discussion towards the general term, using common parameters.o Introduce students to these parameters:

o a, n and d, find tn and the formula for the general term.

A variety of examples involving arithmetic sequences should be given to students (including word problems).

1. 2, 5, 8, 11, 14, ...find the 100th term and the general term.

2. 9, 2, -5, -12, -18, ...find the 200th term and the general term.

3. -55 in the sequence 26, 23, 20, ... is which term number?


http://www.chinese.new-year.co.uk/calendar.htm

4. A sequence is defined by tn = 3n - 2. Find a and d.

5. In an arithmetic sequence, the 5th term is 20 and the 9th term is 36. Find the common difference, the first term, the general term and the 47th term.

You are going to train for a marathon over the summer holidays. The first week you will run 5 km. Each additional week you run another 2 km. How many kilometres do you run in week 8.

Extension: What is the total distance you will run at the end of eight weeks?

Going Beyond

Group Project: Create your own zodiac calendar with your own animals and year span.

Resources

Math 20-1 (McGraw-Hill: sec 1.1)

http://nrich.maths.org/public/leg.php?group_id=7&code=-64#results

Supporting

Assessment Exit slip – some examples of possible exit slips Give a real-life example of an arithmetic sequence. Given a formula for the general term of an arithmetic sequence, find a parameter

(a, d, n, t, tn).

Glossary








Other


Lesson 2

Arithmetic Series

STAGE 1




Different types of sequences and series exist. We can use mathematics to model the pattern

of the sequence or series.





KNOWLEDGE:


the notation of sequences and series (a, d, n, r, t, tn, Sn)


SKILLS:


create a model for a problem/scenario calculate any specified parameter for a

sequence or series (a, d, n, tn, Sn) find the sum of a sequence or the individual

terms or a series

Lesson Summary

Use a pattern to determine the parameters of arithmetic series. Find the sum of a series.

Lesson Plan

Hook

Visit the following website and discuss the structures in pictures. Discuss how to determine the number of cans in each picture. http://www.canstruction.org/


http://www.canstruction.org/

Lesson

Have students generate three or four different finite arithmetic sequences and find the sum of those sequences.

Teacher Notes: The text resource uses t1 instead of a in the general term and sum formula. Introduce the difference between a finite and infinite sequence. Some students will create short sequences and manually add up the terms. Encourage students to then look for patterns in finding the sum to develop their own

formula.o A prompt could indicate that the last term of the sequence is tn.o It may also help to prompt students to examine a relationship between the number

of terms, the first term and the last term of the sequence in finding the sum of an arithmetic sequence.

Provide a few sequences and the formulae (tn = a + (n - 1)d, Sn =

n2 [2 a+(n−1 ) d ]

and

Sn =

n2 (a+t n)

. Given:1. n, a and d, find the sum2. n, a and tn, find the sum3. Sn, a and n find parameter d4. Sn, a and n find tn

5. Sn, a and d find parameter n6. Sn, n and d find parameter a7. Sn, a and tn find parameter n

Some examples can include:

1. 2, 5, 8, 11, 14, ...find the sum of the first 100 terms.

2. 9, 2, -5, -12, -18, ...find the sum of the first 200 terms.

3. Find the sum of the first 15 terms of the sequence defined by tn = 3n-2.

Relate back to the Canstruction website and have students design a symmetrical shape using soup cans and calculate how many cans would be required to build their shape.

Teacher Notes: For students who finish early, they could work on developing a formula for the sum of

an arithmetic series. Some possible shapes students could design are pyramids, cones, football, etc … For this to work, the number of cans on each level of the shape does have to work out

to be an arithmetic sequence. Students will share their designs with the class.


http://www.canstruction.org/

Going Beyond

Example #2 on p. 26 McGraw-Hill Ryerson Pre-Calculus 11 Textbook

Resources

Math 20-1 (McGraw-Hill: sec 1.1, 1.2)

Supporting

If you wish to relate an arithmetric sequence to a linear function, consider this applet. You may want to use graphing calculators to show the general term of an arithmetic sequence (tn = a + (n - 1)d) as a transformation of the linear function (tn = a + nd). Students may need coaching to realize that what they know about y = mx + b applies to tn = a + nd and ultimately to tn = a + (n - 1)d.

Source: http://www.learnalberta.ca/content/mejhm/html/object_interactives/patterns/explore_it.html

Assessment

Glossary







http://www.learnalberta.ca/content/mejhm/html/object_interactives/patterns/explore_it.html

http://www.learnalberta.ca/content/mejhm/html/object_interactives/patterns/explore_it.html


finite sequence – A sequence that has a specific number of terms.

infinite sequence – A sequence that has an unlimited number of terms.

Other


Lesson 3

Geometric Sequence

STAGE 1

BIG IDEA: . The world is full of patterns to be discovered. Students will be able to recognize a pattern and continue modeling the sequence to make predictions for future elements.









KNOWLEDGE:



the notation of sequences and series (a, d, n, r, t, tn)



SKILLS:





series

Lesson Summary

Develop understanding/formula of the general term of a geometric sequence.


Lesson Plan

Introduction

When Spider Man was bitten, the radioactive spider injected 1 mg of venom into his body. The venom concentration doubles every hour. How many mg were in his blood stream eight hours later?

A car brand new costs $25 000. Each year, on average, the car is worth 80% of the previous year. In what year is it worth half of its original value?

Lesson

Determine the common ratio for a given geometric sequence.

Find specific terms given the general term.

Given:1. n, r, tn, find parameter a2. a, n, tn, find parmeter r

Going Beyond

Example #3, page 36 (method 2) McGraw-Hill Ryerson Pre-Calculus 11 Textbook

Resources



Supporting

Consider showing the trailer of the first Spiderman movie.http://www.youtube.com/watch?v=FN3YaybNJ2s&safety_mode=true&persist_safety_mode=1


http://www.youtube.com/watch?v=FN3YaybNJ2s&safety_mode=true&persist_safety_mode=1


If you wish to show the general term of a geometric sequence as a transformation of the exponential function, consider:

source http://staff.argyll.epsb.ca/jreed/math30p/logarithms/sequence.htm

Assessment

Exit slipAssessed on unit exam

Glossary





Other


http://staff.argyll.epsb.ca/jreed/math30p/logarithms/sequence.htm

Lesson 4

Geometric Series

STAGE 1

BIG IDEA: . The world is full of patterns to be discovered. Students will be able to recognize a pattern and continue modeling the sequence to make predictions for future elements.









KNOWLEDGE:




SKILLS:


calculate any specified parameter for a sequence or series (a, d, n, r, tn, Sn)

find the sum of a sequence or the individual terms or a series

calculate the infinite sum of a convergent series

Lesson Summary

Develop an understanding and uses for the formulae of a geometric series.


Lesson Plan

Introduction Activity

You and your parent agree on a payment plan for you to do your household chores for the next 16 weeks. Your parents, thinking they are so smart, agreed to pay you one penny on the first week and keep doubling the payment each week for 4 months (16 weeks). By the end of the 16 weeks what is the total amount of money your parents have paid you to do your chores.

Investigating Fractals Refer to Page 46 of the MGH-Ryerson textbook on Investigating Fractals. Or refer to Applied Math 30 resources for fractal activities

Lesson

Develop the formula for a geometric series in stages.

Sum = (r n−1 )

64 - 1= 63: Sum of 6 iterations = (26−1 )=63 and Sum of n iterations = (r n−1 )Mathematics 20-1 Sequences and Series of 36

Sum = t 1 (rn−1 )2 + 4 + 8 + 16 + 32 + 64 = 126

Try (r n−1 ) : (26−1 )=63 , but 63 is 126/2.

Sum of 6 iterations = 2 (26−1 )=126 and Sum of n iterations = t 1 (rn−1 )

Sum =

t1 (r n−1 )r−1

2 + 6 + 18 + 54 + 162 + 486 = 728 (each term is 3 times the previous term)

Try t 1 (rn−1 ) : 2 (36−1 )=1456 , but 728 is 1456/2 and 3 – 1 = 2

Sum of 6 iterations =

2 (36−1 )3 - 1

=728

Sum of n iterations = Sn =

t1 (r n−1 )r−1

Test the first 2 examples:1 + 2 + 4 + 8 + 16 + 32 = 63

Sn =

t1 (r n−1 )r−1

=1 (26−1 )

2−1 = 63 ✔

2 + 6 + 18 + 54 + 162 + 486 = 728

Sn =

t1 (r n−1 )r−1

=2 (36−1 )

3−1 = 728 ✔

Give geometric series and have students come up with the sums when given:a, r and na, r, and tn

Given Sn, students must determine the parameters:a, given r, nr, given a, ntn, given a, r

Going Beyond


Resources


Supporting

If you wish to show the general term of a geometric sequence as a transformation of the exponential function and a geometric series as the sum of the underlying sequence:

source: http://staff.argyll.epsb.ca/jreed/math30p/logarithms/series.htm

Assessment

Exit Slip

Glossary








Other


http://staff.argyll.epsb.ca/jreed/math30p/logarithms/series.htm

Lesson 5

Infinite Geometric Series

STAGE 1










KNOWLEDGE:




SKILLS:


create a model for a problem/scenario calculate any specified parameter for a

sequence or series (a, d, n, r, t, tn, Sn) find the sum of a sequence or the individual


series

Lesson Summary

Understand the conditions necessary to determine the sum of an infinite geometric series.

Explain why a geometric series is convergent or divergent


Lesson Plan

Hook

Each student will need a blank white sheet of paper. Have students colour one half of the

sheet and label it

12 . Students will then colour half of the remaining white space, labeling it

14 .

Continue this for

18 ,

116 ,

132 , and

164 .

Ex.Step 1 Step 2 Step 3

Explain to students that they have just modelled a geometric series. Can they write out the first few terms? What is the general equation?

Solution:12+ 1

4+ 1

8+ 1

16+ 1

32+ 1

64, .. .

Sn=

12 [( 1

2 )n−1]

12−1

=1−( 12 )

n

Consider asking students to calculate S5 using the information on their coloured sheet as well as by the formula. What would happen if we continued colouring half of the remaining white space an infinite number of times? What sum are we approaching? How does our colour sheet help us check the last answer?

Discuss with students the idea that the sum gets closer and closer to 1. Because the series approaches one value, we say that it is convergent. Consider having students graph

y=1−( 12 )

n

so they can see an example of a convergent graph.


What would happen if we coloured twice as much each time?12 + 1 + 2 + 4 + 8 + …

Sn=

12 [ (2 )n−1 ]

2−1=(2 )n−1−1

2

Does this series approach one particular sum?

Explain that series that do not approach a particular sum are called divergent. Consider

having students graph y= (2 )n−1−1

2 so they can see an example of a divergent graph. See if students can determine the factors, which determine whether a series is convergent (| r | < 1) and divergent (| r | > 1).

Note: If students have not been introduced to absolute value notation, consider writing

| r | < 1 as -1 < r < 1. Consider coming back to this example when students study absolute value. Graphing | r | < 1 and discussing -1 < r < 1 may help students remember the meaning of absolute value.

Introduction Activity

Find the sum of the following series

0 .3=0.3 + 0.03 + 0.003 + 0.0003 +…. 2 + 4 + 8 + 16 +….

Discuss why you can find the sum of the first example but not the second?

Distinguish between divergent and convergent series.

Lesson

Math 20-1 (McGraw-Hill, page 60): Convergent Series, and Divergent Series (with or without graphing calculator).

Analyze a geometric sequence to determine whether or not it has a sum. Find the sum of a convergent series.

Math 20-1 (McGraw-Hill, page 60): Infinite Geometric Series, See Math 20-1 (McGraw-Hill, page 64), #11).

For what values of x will this series become convergent?

Note:


At this point students may be able to follow the steps of the examples using S∞=( t1

1−r ), but

have trouble applying what they know to new examples. After students complete Math 20-1

(McGraw-Hill, page 64), #11), consider returning to 0 . 3 . Rewrite the series, using the first term as a common factor. 0.3(1 + 0.1 + 0.01 + 0.001 +….)

Now 1 becomes t1 and t2 is r in the new series. S∞=0 .3( t 1

1−r )=0. 3( 11−0.1 ) .

Help students understand that when the first term is factored outS∞=( t1

1−r ) becomes

S∞=t1( 11−r )

, where t1 = 1 and t2 = r.

Going Beyond

Resources


Supporting

Assessment

Glossary

convergent series – A series in which the sum is finite, where r is between -1 and +1

divergent series – A series that is not convergent, where r is greater than or equal to 1 or less than or equal to -1.




Other


english language arts 10 – 2 web viewthere are 41 home games in the regular season. ......

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