MATHEMATICS 20-1
Sequences and Series
High School collaborative venture with
Edmonton 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 Trimble
Harry Ainlay: Ben Luchkow
Harry Ainlay: Darwin Holt
Harry Ainlay: Lareina Rezewski
Harry Ainlay: Mike Shrimpton
J. Percy Page: Debbie Younger
Jasper Place: Matt Kates
Jasper Place: Sue Dvorack
Millwoods Christian: Patrick Ypma
Ross Sheppard: Patricia Elder
Ross Sheppard: Dean Walls
W. P. Wagner: Amber Steinhauer
M. E. LaZerte: Teena Woudstra
Queen Elizabeth: David Underwood
Strathcona: 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
Page 2 of 35
TABLE OF CONTENTS
STAGE 1
DESIRED RESULTS
PAGE
Big Idea
4
Enduring Understandings
4
Essential Questions
5
Knowledge
6
Skills
7
STAGE 2
ASSESSMENT EVIDENCE
Transfer Task
Arena Plan
Teacher Notes for Transfer Task and Rubric
Transfer Task
Rubric
Possible Solution
8
10
12
14
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
Mathematics 20-1
Sequences and Series
Page 3 of 35
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.
Implementation note:
Post the BIG IDEA in a prominent
place in your classroom and refer to
it often.
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.
Mathematics 20-1
Sequences and Series
Page 4 of 35
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?
Mathematics 20-1
Sequences and Series
Page 5 of 35
Knowledge:
Enduring
Understanding
Specific
Outcomes
Students will know …
Students will understand…

Different types of
sequences and series
exist.
Description of
Knowledge
*RF9
RF10




Students will know …
Students will understand…

We can use
mathematics to model
the pattern of the
sequence or series.
RF9
RF10




8888
I*RF =
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
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
Relations and Functions
Mathematics 20-1
Sequences and Series
Page 6 of 35
Skills:
Enduring
Understanding
Specific
Outcomes
Students will be able to…
Students will understand…

Different types of
sequences and series
exist.
Description of
Skills
* RF9
RF10




Students will be able to…
Students will understand…

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
We can use
mathematics to model
the pattern of the
sequence or series.





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
calculate the infinite sum of a convergent
series
* RF = Relations and Functions
Implementation note:
Teachers need to continually ask
themselves, if their students are
acquiring the knowledge and skills
needed for the unit.
Mathematics 20-1
Sequences and Series
Page 7 of 35
STAGE 2
1
Assessment Evidence
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.
Implementation note:
Teachers need to consider what performances and
products will reveal evidence of understanding?
What other evidence will be collected to reflect
the desired results?
Mathematics 20-1
Sequences and Series
Page 8 of 35
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.
Mathematics 20-1
Sequences and Series
Page 9 of 35
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.
b.
c.
d.
e.
The number of seats in the first row.
The number of rows required.
The number of seats by which each row increases.
The number of seats in the last row.
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
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.
Assessment
Mathematics 20-1
Sequences and Series
Rubric
Level
Excellent
Criteria
4
Math
All required
Content elements are
Part 1
present and
correct
Math
Content
Part 2
All required
elements are
present and
correct
Math
Content
Part 3
All required
elements are
present and
correct
Present
s Data
Presentation of
data is clear,
precise and
accurate
Explains Provides
Choices insightful
explanations
Proficient
3
All required
elements are
present but
may contain
minor errors
All required
elements are
present but
may contain
minor errors
All required
elements are
present but
may contain
minor errors
Presentation of
data is
complete and
unambiguous
Provides
logical
explanations
Adequate
2
Some required
elements are
missing, or
contain major
errors
Some required
elements are
missing, or
contain major
errors
Some required
elements are
missing, or
contain major
errors
Presentation of
data is
simplistic and
plausible
Provides
explanations
that are
complete but
vague
Limited
1
Most required
elements are
missing or
incorrect
Insufficient
Blank
No score is awarded
as there is no
evidence given
Most required
elements are
missing or
incorrect
No score is awarded
as there is no
evidence given
Most required
elements are
missing or
incorrect
No score is awarded
as there is no
evidence given
Presentation of
data is vague
and
inaccurate
Provides
explanations
that are
incomplete or
confusing.
Presentation of data
is
incomprehensible
No explanation is
provided
When work is judged to be limited or insufficient, the teacher makes decisions.
Mathematics 20-1
Sequences and Series
Page 13 of 35
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.
Sn =
(
)
n éë2a + n - 1 d ùû
2
40 éë2 460 + 40 - 1 4ùû
=
2
= 21520
(
) (
)
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.
t n = ar n -1
(
)
= 400 0.92
= 15.48
40-1
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.
Mathematics 20-1
Sequences and Series
Page 14 of 35
Row
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
Seats
$/Gm
460
$400.00
464
$368.00
468
$338.56
472
$311.48
476
$286.56
480
$263.63
484
$242.54
488
$223.14
492
$205.29
496
$188.86
500
$173.76
504
$159.85
508
$147.07
512
$135.30
516
$124.48
520
$114.52
524
$105.36
528
$96.93
532
$89.17
536
$82.04
540
$75.48
544
$69.44
548
$63.88
552
$58.77
556
$54.07
560
$49.75
564
$45.77
568
$42.10
572
$38.74
576
$35.64
580
$32.79
584
$30.16
588
$27.75
592
$25.53
596
$23.49
600
$21.61
604
$19.88
608
$18.29
612
$16.83
616
$15.48
Mathematics 20-1
Row's
Row's
Revenue/GM
Games
Revenue/Season
$184,000.00
41
$7,544,000.00
$170,752.00
41
$7,000,832.00
$158,446.08
41
$6,496,289.28
$147,016.29
41
$6,027,668.07
$136,401.22
41
$5,592,450.00
$126,543.65
41
$5,188,289.75
$117,390.33
41
$4,813,003.46
$108,891.66
41
$4,464,557.92
$101,001.47
41
$4,141,060.44
$93,676.81
41
$3,840,749.39
$86,877.69
41
$3,561,985.32
$80,566.90
41
$3,303,242.71
$74,709.81
41
$3,063,102.21
$69,274.23
41
$2,840,243.43
$64,230.20
41
$2,633,438.21
$59,549.86
41
$2,441,544.26
$55,207.30
41
$2,263,499.34
$51,178.43
41
$2,098,315.73
$47,440.86
41
$1,945,075.09
$43,973.75
41
$1,802,923.74
$40,757.76
41
$1,671,068.12
$37,774.89
41
$1,548,770.69
$35,008.44
41
$1,435,346.02
$32,442.86
41
$1,330,157.15
$30,063.71
41
$1,232,612.30
$27,857.60
41
$1,142,161.61
$25,812.06
41
$1,058,294.31
$23,915.51
41
$980,535.95
$22,157.22
41
$908,445.84
$20,527.19
41
$841,614.72
$19,016.16
41
$779,662.53
$17,615.52
41
$722,236.35
$16,317.28
41
$669,008.52
$15,114.02
41
$619,674.83
$13,998.85
41
$573,952.88
$12,965.38
41
$531,580.52
$12,007.67
41
$492,314.44
$11,120.21
41
$455,928.81
$10,297.90
41
$422,214.08
$9,535.99
41
$390,975.75
Sequences and Series
Page 15 of 35
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:
ESSENTIAL QUESTIONS:
Students will understand …


We can use mathematics to model the pattern
of the sequence or series.


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?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …




the components required to finding the
general term
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)
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.
Mathematics 20-1
Sequences and Series
Page 16 of 35
Lesson Summary

Use patterns to generate the general term of an arithmetic sequence and examine
each parameter.
Lesson Plan
Think
Why do we need/where do we see sequences and series?
Explore Patterns
Give examples of arithmetic, geometric and other sequences. Ask students to find the next
two terms and determine a rule for the pattern.
Hook
What 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?
4. A sequence is defined by tn = 3n - 2. Find a and d.
Mathematics 20-1
Sequences and Series
Page 17 of 35
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
arithmetic sequence - A sequence for which the difference between successive terms is
constant
common difference - A constant that is added to each term to produce an arithmetic
sequence
Mathematics 20-1
Sequences and Series
Page 18 of 35
first term – The first value in a/an arithmetic/geometric sequence/series.
general term - A function that describes all terms in a sequence
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.
Other
Mathematics 20-1
Sequences and Series
Page 19 of 35
Lesson 2
Arithmetic 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.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …



Different types of sequences and series exist.
We can use mathematics to model the pattern
of the sequence or series.


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?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …




the notation of sequences and series (a, d, n,
r, t, tn, Sn)
the components required to finding the
general term

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/
Mathematics 20-1
Sequences and Series
Page 20 of 35
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 =
(
)
n
a + t n . Given:
2
n, a and d, find the sum
n, a and tn, find the sum
Sn, a and n find parameter d
Sn, a and n find tn
Sn, a and d find parameter n
Sn, n and d find parameter a
Sn, a and tn find parameter n
Sn =
1.
2.
3.
4.
5.
6.
7.
né
2a + n - 1 d ùû and
2ë
(
)
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 t n = 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.
Mathematics 20-1
Sequences and Series
Page 21 of 35
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
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
first term – The first value in a/an arithmetic/geometric sequence/series.
general term - A function that describes all terms in a sequence
Mathematics 20-1
Sequences and Series
Page 22 of 35
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.
finite sequence – A sequence that has a specific number of terms.
infinite sequence – A sequence that has an unlimited number of terms.
Other
Mathematics 20-1
Sequences and Series
Page 23 of 35
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.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …



Different types of sequences and series exist.
We can use mathematics to model the pattern
of the sequence or series.


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?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …







the difference between arithmetic and
geometric
the notation of sequences and series (a, d, n,
r, t, tn)
the components required to finding the
general term
the difference between convergent and
divergent geometric series and what leads to
convergence


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
calculate the infinite sum of a convergent
series
Lesson Summary

Develop understanding/formula of the general term of a geometric sequence.
Mathematics 20-1
Sequences and Series
Page 24 of 35
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 a
2. a, n, tn, find parmeter r
Going Beyond
Example #3, page 36 (method 2) McGraw-Hill Ryerson Pre-Calculus 11 Textbook
Resources
Math 20-1 (McGraw-Hill: sec 1.3)
http://nrich.maths.org/public/leg.php?group_id=7&code=-64#results
Supporting
Consider showing the trailer of the first Spiderman movie.
http://www.youtube.com/watch?v=FN3YaybNJ2s&safety_mode=true&persist_safety_mode=1
Mathematics 20-1
Sequences and Series
Page 25 of 35
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 slip
Assessed on unit exam
Glossary
common ratio - A constant that is multiplied to each term to produce a geometric sequence
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
Other
Mathematics 20-1
Sequences and Series
Page 26 of 35
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.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …



Different types of sequences and series exist.
We can use mathematics to model the pattern
of the sequence or series.


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?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …



the notation of sequences and series (a, d, n,
r, t, tn)
the components required to finding the
general term


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.
Mathematics 20-1
Sequences and Series
Page 27 of 35
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
Page 28 of 35
(
)
Sum = t1 r n - 1
2 + 4 + 8 + 16 + 32 + 64 = 126
(
) (2 - 1) = 63 , but 63 is 126/2.
Sum of 6 iterations = 2 ( 2 -1) = 126 and Sum of n iterations = t ( r
Try r n - 1 :
6
6
n
1
Sum =
(
)
t1 r n - 1
r -1
2 + 6 + 18 + 54 + 162 + 486 = 728
(
) (
)
(each term is 3 times the previous term)
Try t1 r n - 1 : 2 36 - 1 = 1456 , but 728 is 1456/2 and 3 – 1 = 2
Sum of 6 iterations =
(
) = 728
2 36 - 1
3-1
t1 r n - 1
Sum of n iterations = Sn =
r -1
(
)
Test the first 2 examples:
1 + 2 + 4 + 8 + 16 + 32 = 63
t1 r n - 1 1 26 - 1
Sn =
= 63 
=
r -1
2 -1
(
) (
)
2 + 6 + 18 + 54 + 162 + 486 = 728
t1 r n - 1 2 36 - 1
Sn =
= 728 
=
r -1
3 -1
(


)
-1
) (
)
Give geometric series and have students come up with the sums when given:
a, r and n
a, r, and tn
Given Sn, students must determine the parameters:
a, given r, n
r, given a, n
tn, given a, r
Going Beyond
Mathematics 20-1
Sequences and Series
Page 29 of 35
Resources
Math 20-1 (McGraw-Hill: sec 1.4)
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
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
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
Other
Mathematics 20-1
Sequences and Series
Page 30 of 35
Lesson 5
Infinite 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.
ENDURING UNDERSTANDINGS:
ESSENTIAL QUESTIONS:
Students will understand …



Different types of sequences and series exist.
We can use mathematics to model the pattern
of the sequence or series.


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?
KNOWLEDGE:
SKILLS:
Students will know …
Students will be able to …




the notation of sequences and series (a, d, n,
r, t, tn)
the components required to finding the
general term


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
terms or a series
calculate the infinite sum of a convergent
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
Mathematics 20-1
Sequences and Series
Page 31 of 35
Lesson Plan
Hook
Each student will need a blank white sheet of paper. Have students colour one half of the
1
1
sheet and label it . Students will then colour half of the remaining white space, labeling it .
2
4
1 1 1
1
Continue this for ,
,
, and
.
8 16 32
64
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:
1 1 1 1
1
1
+ + +
+
+ , ...
2 4 8 16 32 64
ù
1 éæ 1 ö
êç ÷ - 1ú
n
2 êè 2 ø
úû
æ 1ö
ë
Sn =
= 1- ç ÷
1
è 2ø
-1
2
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?
n
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
n
æ 1ö
y = 1- ç ÷ so they can see an example of a convergent graph.
è 2ø
What would happen if we coloured twice as much each time?
Mathematics 20-1
Sequences and Series
Page 32 of 35
1
+1+2+4+8+…
2
1é n ù
2 -1
n-1
1
ê
úû
Sn = 2 ë
= 2 2 -1
2
()
()
Does this series approach one particular sum?
Explain that series that do not approach a particular sum are called divergent. Consider
n-1
1
- so they can see an example of a divergent graph. See if
having students graph y = 2
2
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:
Mathematics 20-1
Sequences and Series
Page 33 of 35
æ t ö
At this point students may be able to follow the steps of the examples using S¥ = ç 1 ÷ , but
è 1- r ø
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 +….)
æ t ö
æ 1 ö
Now 1 becomes t1 and t2 is r in the new series. S¥ = 0.3 ç 1 ÷ = 0.3 ç
.
è 1- 0.1÷ø
è 1- r ø
æ t ö
Help students understand that when the first term is factored out S¥ = ç 1 ÷ becomes
è 1- r ø
æ 1 ö
, where t1 = 1 and t2 = r.
S¥ = t1 ç
è 1- r ÷ø
Going Beyond
Resources
Math 20-1 (McGraw-Hill: sec 1.5)
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.
infinite sequence - A sequence that does not end or have a final term
Mathematics 20-1
Sequences and Series
Page 34 of 35
infinite geometric series – A geometric series that does not end or have a final term. An
infinite geometric series may be convergent or divergent.
Other
Mathematics 20-1
Sequences and Series
Page 35 of 35