ArithmeticSequences

ArithmeticSequences
Teaching & Learning Plans
Arithmetic Sequences
Leaving Certificate Syllabus
The Teaching & Learning Plans
are structured as follows:
Aims outline what the lesson, or series of lessons, hopes to achieve.
Prior Knowledge points to relevant knowledge students may already have and also
to knowledge which may be necessary in order to support them in accessing this new
topic.
Learning Outcomes outline what a student will be able to do, know and understand
having completed the topic.
Relationship to Syllabus refers to the relevant section of either the Junior and/or
Leaving Certificate Syllabus.
Resources Required lists the resources which will be needed in the teaching and
learning of a particular topic.
Introducing the topic (in some plans only) outlines an approach to introducing the
topic.
Lesson Interaction is set out under four sub-headings:
i.
Student Learning Tasks – Teacher Input: This section focuses on possible lines
of inquiry and gives details of the key student tasks and teacher questions which
move the lesson forward.
ii.
Student Activities – Possible Responses: Gives details of possible student
reactions and responses and possible misconceptions students may have.
iii. Teacher’s Support and Actions: Gives details of teacher actions designed to
support and scaffold student learning.
iv. Assessing the Learning: Suggests questions a teacher might ask to evaluate
whether the goals/learning outcomes are being/have been achieved. This
evaluation will inform and direct the teaching and learning activities of the next
class(es).
Student Activities linked to the lesson(s) are provided at the end of each plan.
Teaching & Learning Plan:
Leaving Certificate Syllabus
Aims
• To understand the concept of arithmetic sequences
• To use and manipulate the appropriate formulas
• To apply the knowledge of arithmetic sequences in a variety of contexts
Prior Knowledge
Students have prior knowledge of:
• Patterns
• Basic number systems
• Sequences
• Ability to complete tables
• Basic graphs in the co-ordinate plane
• Simultaneous equations with 2 unknowns
• The nth term (Tn) of an arithmetic sequence
Learning Outcomes
As a result of studying this topic, students will be able to:
• recognise arithmetic sequences in a variety of contexts
• recognise sequences that are not arithmetic
• apply their knowledge of arithmetic sequences in a variety of contexts
• apply the relevant formula in both theoretical and practical contexts
• calculate the value of the first term (a), the common difference (d) and the
general term (Tn) of an arithmetic sequence from information given about
the sequence
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1
Teaching & Learning Plan: Arithmetic Sequences
Catering for Learner Diversity
In class, the needs of all students, whatever their level of ability level, are equally
important. In daily classroom teaching, teachers can cater for different abilities by
providing students with different activities and assignments graded according to levels
of difficulty so that students can work on exercises that match their progress in learning.
Less able students may engage with the activities in a relatively straightforward way
while the more able students should engage in more open-ended and challenging
activities. In this Teaching and Learning Plan, for example teachers can provide students
with different applications of arithmetic sequences and with appropriate amounts and
styles of support.
In interacting with the whole class, teachers can make adjustments to suit the needs
of students. For example, the Fibonacci sequence can be presented as a more
challenging topic for some students.
Apart from whole-class teaching, teachers can utilise pair and group work to encourage
peer interaction and to facilitate discussion. The use of different grouping arrangements
in these lessons should help ensure that the needs of all students are met and that
students are encouraged to verbalise their mathematics openly and share their learning.
Relationship to Leaving Certificate Syllabus
Students
Students
learn about working at FL
should be able
to
In addition,
students working
at OL should be
able to
3.1 Number
systems
–– generalise and
explain patterns
and relationships
in algebraic form
–– appreciate
that processes
can generate
sequences of
numbers or
objects
–– investigate
patterns
among these
sequences
In addition,
students working
at HL should be
able to
–– recognise
whether a
sequence is
arithmetic,
geometric or
neither
–– use patterns to –– find the sum to
continue the
n terms of an
sequence
arithmetic series
–– generate rules/
formulae from
those patterns
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2
Teaching & Learning Plan: Arithmetic Sequences
Lesson Interaction
Student Learning
Tasks: Teacher Input
Student Activities: Possible Responses
Teacher’s Support and
Actions
Assessing the Learning
Teacher Reflections
Section A: To introduce arithmetic sequences or arithmetic progressions
and gain an understanding of the formula Tn= a + (n - 1) d
»» Can you give me
• Blue, red, blue, red...»
examples of patterns
you have already
• 2, 3, 5, 8, 12...»
encountered?
• 1, 4, 9, 16, 25...»
»» Revise the concept of
pattern as dealt with
at Junior Certificate
level.
»» Are the students
familiar with patterns?»
»» Have students come
up with examples of
different patterns?
• 3, 8, 13, 18, 23...
»» Can you now do
questions 1 - 4 on
Section A: Student
Activity 1?
© Project Maths Development Team 2012
• Students should try out these questions,
compare answers around the class and
have a discussion about why their answers
do not all agree.
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»» Distribute Section A:
Student Activity 1 to
the students.»
»» Can students see that
these problems have a
pattern?»
»» Give students time to
explore and to discuss
what is happening.
»» Do students recognise
that any term (apart
from the first term) in
the various patterns
is formed by adding a
fixed number to the
preceding term?
KEY: » next step
• student answer/response
3
Teaching & Learning Plan: Arithmetic Sequences
Student Learning Tasks: Teacher Input
Student Activities: Possible
Responses
»» What have the first three questions
in Section A: Student Activity 1 got in
common?»
»
»
• For each question we were
given an initial value and a
value that was added to each
previous term to generate the
next term.
»» Sequences where you are given an initial
term and where each subsequent term is
found by adding a fixed number to the
previous term are known as an arithmetic
sequences or arithmetic progressions (APs).»
Teacher’s Support and
Actions
Assessing
the Learning
Teacher Reflections
»» Write the words
'arithmetic sequence'
on the board.»
»
»
»
»» The initial term is denoted by a and d is
used to denote the common difference. d
is the number that is added to each term to
generate the next term.»
»
»
»» Write a = Initial term
(First term) and d =
number added to
each consecutive term
(common difference) on
the board.
»» This notation is used by mathematicians.
»» We call the 4th term T4, the ninth term T9
and the nth term Tn.
»
»
»» In each of the three problems on the
Section A: Student Activity 1, ask how we
got each term?
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»» Write
• 4th term =T4
• 9th term =T9
• nth term =Tn on the
board.
• We were given an initial term
a and the next term was found
by adding d (the common
difference) to this initial term
and then the 3rd term was
found by adding d to the
second term.
KEY: » next step
• student answer/response
4
Teaching & Learning Plan: Arithmetic Sequences
Student Learning Tasks:
Teacher Input
Student Activities: Possible
Responses
Teacher’s Support and Actions
Assessing the
Learning
»» Now let’s see if we can
come up with a formula
for Tn?
»
»
• Students should try this
themselves and compare
answers around the class and
have a discussion about why
the answers do not all agree.»
»» Give students time to explore
and discuss what is happening.»
»» Were students able
to come up with
the first 5 terms?»
»» Start with T1. What is T1
in terms of a and d?
•
T1 = a
»
»» Then move on to T2. If we
know what T1 is, how do
we find T2?
• We add ‘d’ to T1 to get T2 so T2
=a+d
»
»» Write out the next three
terms.»
»
•
»» Do you notice a pattern
occurring?»
»
T3= T2 + d = a + d + d = a + 2d
T4= T3 + d = a + 2d + d = a +3d Note: Explain that this formula only
T5= T4 + d = a + 3d + d = a + 4d applies to an arithmetic sequence
that d always has to be a constant
• Yes, the number of ds is always and a is the first term.
one less than the term we are
looking for.»
»» So what is T8?
•
T8= a + 7d
»» What is T38?
•
T38= a + 37d
»» What is T107?
•
T107 = a + 106d
»» What is Tn?
•
Tn= a + (n - 1)d
© Project Maths Development Team 2012
»» Write the following on the
board as students come up with
the terms.
T1 = a
T2 = a + d
T3 = T2 + d = a + d + d = a + 2d
T4 = T3 + d = a + 2d + d = a + 3d
T5 = T4 + d = a + 3d + d = a + 4d
...
Tn = a + (n - 1) d
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Teacher Reflections
»» Do students
recognise that
for a particular
sequence, d always
has the same
value?»
»» Do students
recognise that the
8th term is
T8 = a + (8 - 1)d,
T38 = a + (38 - 1)d,
T107 = a + (106)d,
before getting Tn?
»» Do students
understand the
meanings of a, d
and Tn= a + (n - 1)
d?
KEY: » next step
• student answer/response
5
Teaching & Learning Plan: Arithmetic Sequences
Student Learning Tasks:
Teacher Input
Student Activities: Possible
Responses
»» Continue with Section A:
Student Activity 1.
Teacher’s Support and Actions
Assessing the
Learning
Note: Students will need to understand
that a and d may be fractions, decimals
or negative numbers. They should be
allowed to discover this rather than
being told at the beginning of the
activity sheet. They should also recognise
that n is always a positive integer.
»» Are students
capable of
answering
the questions
on Section A:
Student Activity
1?
Teacher Reflections
»» Questions on Section A: Student
Activity 1 can be done incrementally
and answers discussed with the class.
Reflection:
»» What is an arithmetic
sequence?»
»
»
»
»
»» Conduct a discussion on what has
been learned to date on arithmetic
sequences.»
»» What is n?
• An arithmetic sequence is one
in which each term, apart from
the first term, is generated by »» More time may have to be spent on
adding a fixed number to the
the questions in Section A: Student
preceding term. »
Activity 1 should the rate of student
progress require it.»
• a is the first term
»» Having cognisance of the students’
• d is the common difference
abilities, select which homework
questions on this activity sheet are to
• n is the number of terms
be completed.
»» What is the meaning of Tn?
•
Tn is the nth term
»» What is the formula for Tn?
•
Tn = a + (n - 1) d
»» What is a?
»» What is d?
»» Are students
familiar with the
notation a, d, n
and Tn?
»» Can students use
the formula »
Tn = a + (n - 1) d?
»» For homework, complete
the following questions (pick
suitable questions) from the
Section A: Student Activity 1.
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KEY: » next step
• student answer/response
6
Teaching & Learning Plan: Arithmetic Sequences
Student Learning Tasks: Teacher
Input
Student Activities: Possible
Responses
Teacher’s Support and
Actions
Assessing the
Learning
Section B: To further explore the concepts of arithmetic sequence and
attempt more difficult exercises
»» Class, I would like you to divide
into groups of 2 (or 3) and on
a sheet of paper write a list
of examples / applications of
where arithmetic sequences
occur in everyday life. Justify
why your examples are
arithmetic sequences.
• Saving regular amounts.»
• Depreciation or inflation by
regular amounts.»
• Spending regular amounts.»
• Heights of buildings where each
floor above the ground floor is
the same height.»
• Removing regular equal
amounts from a container.
Note: Students need to able
to come up with examples of
arithmetic sequences other
than those they already met
in class. They should also be
able to recognise, with their
teacher's help, what makes
the sequences they chose
arithmetic. Add examples
of sequences that are not
arithmetic and discuss these
also.»
Teacher Reflections
»» Are students
capable of
justifying that
the sequences
they chose are
arithmetic?
»» Give students time to
discuss their examples.
»» Let’s look again at the formula
Tn = a + (n - 1) d.
»» Distribute Section B:
Student Activity 2.
»» Complete Section B: Student
Activity 2.
»» Select questions depending
on students’ progress
and abilities. Include key
questions 8 and 12-14. »
»» Give students an
opportunity to attempt
questions before the
solutions are demonstrated
on the board.
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KEY: » next step
»» Are students
capable of
applying the
formula »
Tn = a + (n - 1) d?
»» Are students
capable of
developing
stories from
graphs?
• student answer/response
7
Teaching & Learning Plan: Arithmetic Sequences
Student Learning Tasks: Teacher Input
Student Activities: Possible
Responses
Teacher’s Support and
Actions
Assessing the
Learning
Section C: To enable students understand how simultaneous equations
can be used to solve problems involving arithmetic sequences
»» Solve this problem: “Mark’s savings
pattern obeys an arithmetic sequence.
On the 4th week he saved €9 and on
the 6th week he saved €13.
»» Design 2 equations in terms of a and d
to represent Mark’s savings pattern.»
•
»» How do we solve these equations?»
»
»
»
»
»
»
»
»
• Students should come
up with the concept of
simultaneous equations.»
a + 5d = 13
a + 3d = 9
2d = 4
d=2
a + 5(2)= 13
a=3
»» What does a = 3 and d = 2 mean in the
context of the question?»
»
»
»» How much will he save in the 10th
week?
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a + 3d = 9
a + 5d = 13
• Mark saved €3 in the first
week and increased the
amount he saved each
week by €2.
•
T10 = 3 + (10 - 1) 2
T10 = 21
»» Ask an individual student
to write the solution on
the board and explain
what he or she is doing in
each step.»
»» Can students
recognise
what a set of
simultaneous
equations are?»
Note: Students should
already be familiar with
simultaneous equations,
but may need a quick
revision exercise on how
to solve them. It should
be emphasised that
simultaneous equations can
be represented by letters
other than x and y.
»» Do students
recognise that the
a and the d have
the same value in
both equations?»
Teacher Reflections
»» Can students
solve the
simultaneous
equations?
»» The time spent on this
session will depend on
the students’ abilities
and the understanding
they already have of
simultaneous equations.»
»» Another example may
need to be used here if
students are unfamiliar
with this concept.
KEY: » next step
• student answer/response
8
Teaching & Learning Plan: Arithmetic Sequences
Student Learning Tasks:
Teacher Input
Student Activities: Possible
Responses
»» Complete Section C: Student
Activity 3.
Note: Pick the appropriate
questions for the class, reserving
some for homework and exclude
numbers 10-12. At this point.
»» I would like you to work in
• Students exchange their
pairs and devise your own
equations.»
question(s) similar to the ones
in this Student Activity. Then
• Students offer their
give the questions to the
solutions and explain how
group next to you so that they
they arrived at them.
can solve them.
Teacher’s Support
and Actions
Assessing the Learning
»» Distribute Section C: »» Are students able to
Student Activity 3.
establish and solve the
sets of simultaneous
Note: For question
equations that represent
No 7, weight loss
most of the problems
is normally not a
in Section C: Student
constant loss per
Activity 3?
month hence weight
loss is not a good
example of an
arithmetic sequence.
Teacher Reflections
»» Write a variety
»» Are students capable
of simultaneous
of devising their own
equations from the
questions?
students on the
board together with
their solutions. »
»» Allow students to
talk through their
work so that any
misconceptions
become apparent
and are dealt with.
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KEY: » next step
• student answer/response
9
Teaching & Learning Plan: Arithmetic Sequences
Student Learning Tasks: Teacher Input
Student Activities:
Possible Responses
Teacher’s Support
and Actions
Assessing the
Learning
Section D: To enable students deal with problems like the following:
»» A mobile phone supplier charges €5 plus €0.10
per call.»
»» What is the charge when the user makes 8 calls?»
• €5.80
»» Which term of the sequence is this?»
• 9th»
»» Does the cost of using this supplier form an
arithmetic sequence?»
• Yes»
»
»» Why do you say it forms an arithmetic sequence?»
»
»
»
• Because you add 0.10
for every call made,
so it has the common
difference.»
»» What value has a in this sequence?
• 5»
»» What value has d in this sequence?
• 0.10»
»» Which term in the sequence represents the 10th
call? »
»» This is something you need to watch out for in
questions.»
»» Explain why it happened in this question?»
»» Now attempt questions 10 -12 in Section C:
Student Activity 3.
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»» Put the following
costs on the
board:»
€5 plus €0.10 per
call.»
»» Allow students to
talk through their
work so that any
misconceptions
become apparent
and are dealt
with.
Teacher Reflections
»» Can students
distinguish which
formula to use?»
»» Do students
understand why
in this question
the term that
represents the
10th call is the 11th
term?
• 11th term»
»
»
»
»
• Because the initial
term was for no calls
rather than one call.
KEY: » next step
• student answer/response
10
Teaching & Learning Plan: Arithmetic Sequences
Student Learning Tasks: Teacher Student Activities:
Input
Possible Responses
Teacher’s Support and Actions
Assessing the Learning
Section D: To enable students see that not all sequences are arithmetic sequences.
Note: the Fibonacci sequence is not specified in the syllabus, but is a famous mathematical sequence and a class of more able
students may enjoy exploring this sequence. However, at this stage in the T&L teachers need to challenge their students to
provide examples of some sequences that are not arithmetic.
»» Are all sequences arithmetic?
Explain your reason.»
»» There is a famous sequence
called the Fibonacci Sequence.
See Section D: Student
Activity 4.
• No. To be an arithmetic
sequence Tn - Tn -1 must
be a constant for all the
terms.»
»
»
»
»» Distribute Section D: Student
Activity 4.
Note: Some students may
need individual help with the
explanation of how the sequence
develops. (See the different
rabbit faces.)»
Teacher Reflections
»» Are students able
to identify the
characteristics of an
arithmetic sequence?»
»» Are students able to
derive the terms of the
Fibonacci Sequence?»
»» Is this an arithmetic sequence? » • No.
»» Can you give examples of
other sequences that are not
arithmetic?
»» Delay telling students that this
is not an arithmetic sequence.
Allow students to explore this
for themselves.»
»» Have students come
up with examples of
sequences that are not
arithmetic?
Note: While not specifically
related to this Teaching and
Learning Plan, this section is
important for two reasons (i)
the Fibonacci Sequence is an
important mathematical sequence
and (ii) it is very important that
students are reminded that not
all sequences are arithmetic.
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KEY: » next step
• student answer/response
11
Teaching & Learning Plan: Arithmetic Sequences
Student Learning Tasks:
Teacher Input
Student Activities: Possible
Responses
Teacher’s Support and Actions
»» Class, complete questions
1-6 on Section D: Student
Activity 4 on the Fibonacci
Number Sequence.
•
•
•
•
•
•
Note: The Fibonacci sequence is {0, 1, 1, 2,
3, 5, 8, 13, , , }, hence the explanation of
no rabbits in the first month.
»» Can you see a pattern
developing?
• If we add together any two
consecutive terms we get the
next term.
1
2
3
5
8
13
»» Now, I want you all to try
the interactive quiz called
“Arithmetic Sequence
Quiz” on the Student’s CD.
• The difference between any
term and the preceding term is
always the same or Tn - Tn-1 is a
constant for all the terms.»
»» Write down two formula
used in connection with
arithmetic sequences and
what does each letter used
mean?»
•
© Project Maths Development Team 2012
Note: To explain why the first term is
zero: explain that there were no rabbits
in the first month. Then, there was 1
pair in the second month etc. A more
complicated explanation of the Fibonacci
rabbit problem can be found at http://
en.wikipedia.org/wiki/Fibonacci_number
Teacher Reflections
»» Did the
students
recognise
the
pattern?
»» If computers are not available, hard
copies of the quiz can be printed for
the students.
Reflection:
»» Describe an arithmetic
sequence.»
»
»» Write down any questions
you have on this section of
the course?
Assessing
the Learning
»» Circulate and take note of any
questions or difficulties students have
noted and help them answer them.
Tn= a+(n-1)d
Tn - Tn-1 = d (The common
difference)
a = first term
d = common difference
n = the term
Tn = the nth term
Tn-1 = the term before the nth
term.
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KEY: » next step
• student answer/response
12
Teaching & Learning Plan: Arithmetic Sequences
Section A: Student Activity 1
1. A gardener buys a plant that is 12cm in height. Each week after that the
plant grows 10cm. Note: The plant is 12cm high at the beginning of the
first week.
a. What will be the height of the plant at the beginning of the 1st, 2nd, 3rd,
4th and 5th weeks, if it follows the same pattern?
b. Use graph paper to represent the pattern from part a.
c. Explain your reasoning.
d. What two pieces of information were you given initially?
2. The ground floor of a building is
6 metres in height and each floor
above it is 4 metres in height.
a. Taking the ground floor as
floor one. List the height of the
building at each of the first five
floors.
b. Use graph paper to represent the
pattern from part a.
c. Explain your reasoning.
d. What two pieces of information were you initially given?
3. John has €500, and wants to go on a holiday, but does not have
sufficient money. He decides to save €40 per week for a number of
weeks to make up the deficit.
a. Given that he has €500 at the beginning of the first week, how much
will he have saved at the end of the, 2nd, 3rd, 4th and 5th weeks?
b. Use graph paper to represent the pattern from part a.
c. Explain your reasoning.
d. What two pieces of information were you given initially?
4. What have the previous three questions got in common?
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13
Teaching & Learning Plan: Arithmetic Sequences
Section A: Student Activity 1 (continued)
5. Find T10 and T12 for questions 1, 2 and 3 above.
Formula for T10
Value of T12
Formula for T12
Value of T12
Question 1
Question 2
Question 3
6. Find T1, T2, T3, T5, T10, and T100 of an arithmetic sequence where
a = 4 and d = 6.
T1
T2
T3
T5
T10
T100
7. Find T1, T2, T3, T5, T10, and T100 of an arithmetic sequence where
a = 3 and d = -2.
T1
T2
T3
T5
T10
T100
8. Find T1, T2, T3, T5, T10, and T100 of an arithmetic progression where
a = 3 and d = ½.
T1
T2
T3
T5
T10
T100
9. Find the 7th, 8th and 14th term in the arithmetic sequence 2, 6, 10, 14, ...
7th Term
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8th Term
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14th Term
14
Teaching & Learning Plan: Arithmetic Sequences
Section A: Student Activity 1 (continued)
10.A baker has 400kg of flour on the first day of the month and uses no
flour that day. How much will he have at closing time on the 17th day of
the month, if he uses 24kg each day? Show your calculations using the
formula for Tn of an arithmetic sequence.
11.Marjorie is trying to increase her fitness through exercise. The first
day she walks 1500 metres and every day after that she increases
this distance by 110 metres. How far will she be walking on the 12th
day? Show your calculations using the formula for Tn of an arithmetic
sequence.
12.Find the first 5 terms in each of the following sequences and determine
which of the sequences are arithmetic assuming they follow the same
pattern.
Sequence
T1
T2
T3
Tn = n + 4
Tn = 2n
Tn = 6 + n
Tn = 9 +2n
Tn = 2n - 6
T n = n2
T4
T5
Is the sequence an arithmetic
sequence? Explain your answer.
13.On 1st November Joan has 200 sweets in a box. She eats no sweets
the first day and she eats 8 sweets per day from the box thereafter.
On what day of the month will she have 40 sweets left? Show your
calculations using the formula for Tn of an arithmetic sequence.
14.Joan joins a video club. It costs €12 to join the club and any video she
rents will cost €2. Jonathan’ joins a different video club where there is
no initial charge, but it costs €4 to rent a video. Represent these two
situations graphically and in algebraic form. Do either or both plans
follow the pattern of an arithmetic sequence? Explain your reasoning.
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15
Teaching & Learning Plan: Arithmetic Sequences
Section B: Student Activity 2
1. An art collection is valued at €50,000 in 2001 and its value increases by
€4,000 annually.
a. Show that, as time passes, the value of the collection follows an
arithmetic sequence? List the first ten terms of the sequence and give
the first term and common difference.
b. How much will the collection be worth in 2051?
2. A new car was valued at €21,000 on 1st January 2001, and depreciated
in value by a regular amount each year. On 1st January 2008, it had a
value of zero. Show that the value of the car could follow an arithmetic
sequence. How many terms are there in this sequence? By how much
did it depreciate each year? Show your calculations.
3. The sum of the interior angles of a triangle is 180°, of a quadrilateral is
360° and of a pentagon is 540°. Assuming this pattern continues, find
the sum of the interior angles of a dodecagon (12 sides). Show your
calculations. http://www.regentsprep.org/Regents/math/algtrig/ATP2/SequenceWordpractice.htm
4. A restaurant uses square tables and if one person sits on
each side how many people can sit at each table?
a. If two tables are pushed together, how many people can
be seated? Draw a diagram.
b. If three tables are placed together in a straight line, how
many people can be seated? Draw a diagram.
c. If 4 tables are placed together in a straight line how many people can
be seated?
d. Is the pattern in this question an arithmetic sequence? Explain your
answer.
5. The seats in a theatre are arranged so that each row accommodates
four people less than the previous row. If the first row seats 50 people,
how many will the 9th row seat?
6. The seats in a theatre are arranged so that each row accommodates
four people more than the previous row. If the first row seats 50
people, how many will the 9th row seat?
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16
Teaching & Learning Plan: Arithmetic Sequences
Section B: Student Activity 2 (continued)
7. Find Tn of the following arithmetic sequence 12, 8, 4, ... For what value
of n is Tn = -96?
5
8. a. The pattern represented in the above graph is of
an arithmetic sequence. Explain why this is so.
3
b. What is the 8th term of this sequence?
2
c. Write a story that this pattern could represent.
1
4
0
0
1
2
3
4
5
9. If the Mean Synodic Period of new moons (Period from new moon to
new moon) is approximately 29.53 days. If the first new moon occurred
on the 11th day of 2009, predict the day of the year on which the 5th
new moon occurred in 2009. Show your calculations using appropriate
formula.
10.How many three-digit positive integers exist so that when divided by 7,
they leave a remainder of 5 if the smallest such number is 110 and the
largest is 999? Show your calculations using appropriate formula.
11.If 2k, 5k-1 and 6k+2 are the first 3 terms of an arithmetic sequence, find
k and the 8th term.
12.The sum of three consecutive terms in an arithmetic sequence is 21 and
the product of the two extreme numbers is 45. Find the numbers.
13.Prove that the sequence Tn = 2n + 6 is an arithmetic sequence.
14.Prove that the sequence Tn = n2 + 3 is not an arithmetic sequence.
15.If Marlene saved €40 per week for the first 8 weeks and then saved
€50 per week for the next 8 weeks. Represent this graphically and
explain why her savings pattern for the whole 16 weeks does not form
an arithmetic sequence.
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17
Teaching & Learning Plan: Arithmetic Sequences
Section C: Student Activity 3
(Calculations must be shown in all cases.)
1. The tenth term of an arithmetic sequence is 40 and the twentieth term
is 30. Find the common difference and the first term.
2. A doctor asked a patient to reduce his medication by a regular number
of tablets per week. The reduction plan begins one week after the visit
to the doctor. If 10 weeks after the visit to the doctor the person is
taking 40 tablets and 20 weeks after the visit the person is taking 30.
Find the weekly reduction in his consumption of tablets.
3. Patricia is climbing up a large number of steps and to encourage her on
her way, her friend started counting the number of steps she completes
every minute. From then on she keeps a regular pace. After 2 minutes
she has climbed 56 steps in total and after 8 minutes she has climbed
158 steps in total.
a. Find the number of steps she climbs every minute once her friend
started counting.
b. Find the number of steps she had climbed before her friend started
counting.
If she had not maintained a regular pace, would you still have been
able to use this method?
4. If the 9th floor of a building is 40 metres above the ground and the
ground floor is 4 metres in height and each floor apart from the ground
floor has equal height. Find the height of each floor. (Note the 1st floor
is the one above the ground floor etc.)
5. On the first day of a month a baker receives a delivery of flour and
starts using this flour at the beginning of the following day. He uses a
regular amount of flour each day thereafter. At the end of the 4th day of
the month he has 1,000kg of flour remaining and at the end of the 12th
of the month 640kg remain. How much flour does he use per day and
how much flour did he have delivered? Use equations to represent this
information and then solve the equations.
6. A woman has a starting salary of €20,000 and after the first year she
gets an annual increase of €2,000 per year. Find her annual salary when
she reaches retirement age, 40 years from when she started the job?
© Project Maths Development Team 2012
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18
Teaching & Learning Plan: Arithmetic Sequences
Section C: Student Activity 3 (continued)
7. One week after joining a gym a man starts losing a constant amount
of weight. Eight weeks after joining the gym he weighs 98kg and 15
weeks after joining the gym the weighs 91kgs, find his weight before
joining the gym and the amount of weight he lost each week.
Is a weight loss programme a good application for an arithmetic
6
sequence? Explain your answer.
8. a. Why is the pattern in the following drawing an
arithmetic sequence and what will the length of the
88th side be?
b. List an arithmetic sequence of your choice and draw a
geometric representation of it.
2
5
1
A
3
7
4
G
9. A new company made a profit of €2,000 in the 8th
week of business and in the 12th week of business they made €3,000. If
their accountant informed you that the increases in their profit followed
an arithmetic sequence pattern, how much profit did they make in the
first 4 weeks of business?
10.The membership fee for a gym is €100 and for each visit after that the
charge is €10.
a. How much will ten visits cost? Show your calculations.
b. What will be the cost of joining the gym and visiting it on n occasions?
11.A walker is already walking 5km per day and decides to increase this
amount by 0.1km per day starting on 1st August.
a. What distance will he be walking on 3rd August?
b. Is this an arithmetic sequence? Explain your reasoning.
c. What is the formula for the nth term?
d. If this question had stated that the walker walked 5km on 1st August
and that he had increased this by 0.1km, each day thereafter. What would the formula have been for:
i. The 16th August?
ii. The nth day of the month
12.On 1st June, Joseph has 200 sweets in a box. He eats 8 sweets per day
from the box starting on the 1st June. On what day of the month will he
have 40 sweets left? Show your calculations using algebra. How does
this differ from question 13 in Section A: Student Activity 1?
© Project Maths Development Team 2012
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19
Teaching & Learning Plan: Arithmetic Sequences
Section D: Student Activity 4
Fibonacci was an Italian mathematician, who
worked on this problem in the 13th century.
He studied how fast rabbits breed in ideal
circumstances. He imagined a newly born pair of
rabbits, one male and one female were placed
in a field and made the following assumptions
about the mating habits of rabbits:
• Rabbits mate at exactly one month old and mate every month after that.
• Rabbits always have litters of exactly one male and one female.
• The gestation period for rabbits is exactly one month. (Not true)
• Rabbits never die. (Not true)
1 How many pairs will there be after 1 month? Explain.
2 How many pairs will there be after 2 months? Explain.
3 How many pairs will there be after 3 months? Explain.
4 How many pairs will there be after 4 months? Explain.
5 How many pairs will there be after 5 months? Explain.
6 How many pairs will there be after 6 months? Explain.
7 Can you see a pattern developing? List the first 20 numbers in the
Fibonacci Number Sequence?
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20
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