Mathematics Arithmetic Sequences

BASIC EDUCATION ASSISTANCE FOR MINDANAOLEARNING GUIDE

SECOND YEAR - MATHEMATICS

SEARCHING PATTERNS IN SEQUENCES: ARITHMETIC, GEOMETRIC& OTHERS

Module 12: Arithmetic Sequences: Always Come With A Flow

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Written, edited and produced by Basic Education Assistance for Mindanao, July 2009

BASIC EDUCATION ASSISTANCE FOR MINDANAO


SEARCHING PATTERNS IN SEQUENCES: ARITHMETIC, GEOMETRIC & OTHERS

MODULE 12: ARITHMETIC SEQUENCES: ALWAYS COME WITH A FLOW

INFORMATION ABOUT THIS LEARNING GUIDE

Recommended number of lessons for this Learning Guide: 7

Basic Education Curriculum CompetenciesYear 8 Mathematics: ARITHMETIC SEQUENCES: ALWAYS COME WITH A FLOW Demonstrate knowledge and skill related to arithmetic sequences and apply these in

solving problems.

List the next few terms of a sequence given several consecutive terms.

Derive, by pattern-searching, a mathematical expression (rule) for generating the sequence.

Describe an arithmetic sequence by any of the following ways:

giving the first few terms

giving the formula for the nth term

drawing the graph

given the first few terms of an arithmetic sequence, find the:

common difference

nth term

Given two terms of an arithmetic sequence, find the first term, the common difference or a specified nth term.

Solve problems involving arithmetic means.

Derive the formula for the sum of the n terms of an arithmetic sequence.

Define the sum of an arithmetic sequence.

Solve problems involving arithmetic sequences.

Objectives Determine and draw the next figure given the first three terms.

Form a sequence from the given figures and describe its terms.


Derive, by pattern-searching, a mathematical expression (rule) for generating a sequence.

Describe an arithmetic sequence by: (1) giving the first few terms; (2) giving the formula for the nth term; and (3) drawing the graph.

Find the common difference and the nth term of an arithmetic sequence given the first few terms.

Find the first term, the common difference or a specified nth term given the two terms of an arithmetic sequence.

Derive the formula for the sum of the n terms of an arithmetic sequence.

Define the sum of an arithmetic sequence.

Solve problems involving arithmetic sequence, series and means.

List different series of whole numbers to prove/disprove the given assumption.

Basic Education Assistance for MindanaoLearning Guide, July 2009 3





Essential concepts, knowledge and understandings targeted An arithmetic sequence is any sequence in which each term after the first is obtained by

adding a fixed number d, called the common difference, to the preceding term.

The terms between two given terms in an arithmetic sequence are called arithmetic means. A single arithmetic mean between two numbers is the average, or the arithmetic mean of the two numbers.

A series is an indicated sum of terms of a sequence.

The nth term of an arithmetic sequence is defined as, an = a1 + (n 1)d.

The formula for the sum of an arithmetic sequence is Sn =n2

[2a1 + (n 1)d].

Specific vocabulary introduced arithmetic sequence - a sequence in which each term after the first term is obtained by

adding a constant number called the common difference

arithmetic mean the sum of two terms in a sequence divided by two

arithmetic series the indicated sum of the terms in an arithmetic sequence

common difference the difference between any term and its preceding term in an arithmetic sequence

Suggested organizational strategies Have the classroom ready to accommodate groups of students to explore and learn from

the activities where they will feel at ease and comfortable with their mates.

Assign roles to students within the groups.

Prepare the materials to be used by the students in the different activities.

Prepare also enough and clear copies of the activity sheets prior to the lessons.

Opportunities for IntegrationENGLISH Several activities in this Learning Guide encourage the students to maximize their language

proficiency wherein they will be asked to reason out their common understanding of the concepts with the correct usage of grammar.

PEACE EDUCATION

The activities suggested in this Learning Guide aim to achieve accuracy of the students' outputs while they perform the tasks harmoniously.

VALUES EDUCATION

Students working in a group develop their sense of being sensitive to the needs and feelings of others especially in doing group tasks.

MULTICULTURALISM

The activities in this Learning Guide are suited to different cultures and tribes.

GENDER INCLUSIVITY

The students, boys and girls, are encouraged to share equal responsibilities and participation of the given tasks.






Activities in this Learning GuideActivity 1: UNIQUE PATTERN

Multiple Intelligences

Visual/Spatial

Interpersonal

Logical/Mathematical

Skills

Organization of parts

Use information

Seeing patterns

Activity 2: HIDDEN PATTERN


Visual/Spatial


Skills

Verify the value of evidence

Seeing patterns

Activity 3: REPRESENTED PATTERN


Interpersonal


Visual/Spatial

Skills

Organization of parts

Solve problems using required skills or knowledge

Text Type

Procedural Recount

Activity 4: DISCOVERED PATTERN


Interpersonal


Skills

Knowledge of major ideas






Mastery of subject matter


Activity 5: SUMMED-UP PATTERN



Interpersonal

Verbal/Linguistic

Skills

Understanding information


Verify the value of evidence

Activity 6: DISPLAYED PATTERN



Interpersonal

Verbal/Linguistic

Skills



Activity 7: APPLIED PATTERN



Interpersonal

Skills




Activity 8: UNFOLDED PATTERN



Interpersonal

Verbal/Linguistic






Skills


Seeing patterns



Key Assessment Strategies Games

Puzzles

Performance test

Observing students






Mind MapThe Mind Map displays the organization and relationship between the concepts and activities in this Learning Guide in a visual form. It is included to provide visual clues on the structure of the guide and to provide an opportunity for you, the teacher, to reorganize the guide to suit your particular context.

Stages of LearningThe following stages have been identified as optimal in this unit. It should be noted that the stages do not represent individual lessons. Rather, they are a series of stages over one or more lessons and indicate the suggested steps in the development of the targeted competencies and in the achievement of the stated objectives.

AssessmentAll six Stages of Learning in this Learning Guide may include some advice on possible formative assessment ideas to assist you in determining the effectiveness of that stage on student learning. It can also provide information about whether the learning goals set for that stage have been achieved. Where possible, and if needed, teachers can use the formative assessment tasks for summative assessment purposes i.e as measures of student performance. It is important that your students know what they will be assessed on.






1. Activating Prior LearningThis stage aims to engage or focus the learners by asking them to call to mind what they know about the topic and connect it with their past learning. Activities could involve making personal connections.

Background or purposeIt is beneficial in this stage that the students will be exposed first on the series of figures arranged in a pattern or sequence. They shall determine and draw the next figure of the sequences given the first three terms.

StrategyTIP (Think-Ink-Pair). A strategy that allows individual to reach consensus and check understanding. Students think individually about the problem and write their own understanding. After which, everyone is given a chance to discuss with a partner and record what they have drawn to reach consensus. The activity suggested using this strategy aims to tap the logical/mathematical and visual/spatial intelligences of the students.

Materialactivity sheets on pages 21-22

Activity 1: Unique PatternInstructions:

1. Organize the class into pairs.

2. Hand out the activity sheets to the students.

3. Allow them first to think and work individually the task.

4. After which, let each student discuss his/her output with a partner to reach consensus.

Formative AssessmentCheck the outputs of the students. Refer to page 23 for the answer key.

RoundupThe students should have determined and drawn correctly the fourth figure of the sequences given the first three terms.

2. Setting the ContextThis stage introduces the students to what will happen in the lessons. The teacher sets the objectives/expectations for the learning experience and an overview how the learning experience will fit into the larger scheme.

Background or purposeIn this stage, the students shall form a sequence from the given figures and describe its terms.

StrategySUSTAINED CONVERSATION. This strategy occurs when teachers and students, or students and students discuss issues in a meaningful context. Rather than just asking for a right answer, sustained conversations develop student's reasoning skills, discussion skills and






questions their values and beliefs. This can be used in whole class, small group and individual discussions.

Materials

cut out of figure strips (refer to Teacher Resource Sheet 1 on page 24)

activity sheet on page 25

manila paper

pentel pen

paste/glue

Activity 2: Hidden PatternInstructions:

1. Organize the class into groups of 5 or as desired.

2. Distribute the materials needed.

3. Post the first three figures on the board. Let the students analyze and discover the pattern. Then, encourage them to arrange the strips according to the pattern and paste them on a manila paper.

4. If they cannot guess the correct sequence, cover the other half of each figure with a piece of paper vertically as illustrated below. With this, the students will see that the series of figures are the numbers from 1 to 9 with their symmetries.

Formative AssessmentCheck the outputs of the students.

RoundupThe students should have formed a sequence from the given figures and described its terms.

3. Learning Activity SequenceThis stage provides the information about the topic and the activities for the students. Students should be encouraged to discover their own information.

Background or purposeGiven the different suggested activities in this stage, the students will be expected to fully grasp the concepts on arithmetic sequences. At the end, they will be able to:

list the next few terms of a sequence given several consecutive terms;

derive, by pattern-searching, a mathematical expression (rule) for generating a sequence;

describe an arithmetic sequence by any of the following ways:






- giving the first few terms;

- giving the formula for the nth term; and

- drawing the graph.

find the common difference and the nth term of an arithmetic sequence given the first few terms;

find the first term, the common difference or a specified nth term given the two terms of an arithmetic sequence;

derive the formula for the sum of the n terms of an arithmetic sequence and define its sum; and

solve problems involving arithmetic sequence, series and means.

Strategies INTERACTIVE LECTURE. This strategy provides students with a general outline to give

them a framework for thinking about a subject and to structure their notetaking. This type of lecture involves students by focusing their attention on key concepts. It emphasizes information transfer at the knowledge, recall, and comprehension levels of learning.

COMMUNITY CIRCLE. This is one of the cooperative learning strategies that allows discussions on a certain issue or task. This can be used as a basic tool for group work skills wherein the students are organized into different groups to perform the activity, then discuss and share ideas to gather relevant proofs.

DECODING. A strategy used to translate data or message from a code into the original language or form. In the context of this activity, the students will solve problems on arithmetic sequence. After which, they will look for the corresponding answers on the decoder that will satisfy the given challenge.

Materials

Activity 3


5 boxes of matchsticks

graphing paper

manila paper

pentel pen

Activity 4


Activity 6


Teacher's InputBegin this stage by refocusing students' attention on activity 2. It is expected that the students had discovered the number pattern, 1, 2, 3, 4, 5, 6, 7, 8, 9, out of the figure strips. This time, you may ask them to give the collective name of this number pattern.






A sequence may be finite or infinite depending on its terms. A sequence is said to be finite if it has a first term and a last term. It is infinite if it has a first term but no last term, that is, it is endless.

Activity 3: Represented PatternInstructions:

1. Organize the class into 6 groups.

2. Assign in every group a leader to facilitate the given task.

3. Distribute the materials.

4. Let them finalize their answers on a manila paper.

5. Ask a representative from each group to discuss their output to the class for comparison.

The students should have derived the completed table below.

number of squares (n) 1 2 3 4 5 6 7 8 9 10 n

number of matchsticks (m) 4 7 10 13 16 19 22 25 28 31 3n + 1

The graph of the sequence described in the table is shown below.

The number of matchsticks is derived by using the mathematical sentence, 3n + 1. Therefore, if there will be 100 squares, then the total number of matchsticks will be 301.

In the given challenge, if a1 is the first term, a2 is the second term, and so on and d is the common difference, the equations that will determine the 2nd, 4th, 9th and the nth terms are shown below.

a2 = a1 + d

a3 = (a1 + d) + d or a1 + 2d






a4 = (a1 + 2d) + d or a1 + 3d

a9 = (a1 + 7d) + d or a1 + 8d

This shows that the coefficient of d is one less than the number of terms. Therefore, the nth term will be: an = a1 + (n - 1)d

You may give the following exercises to the students to verify the definition of an arithmetic sequence.

Exercises:

1. Find the 14th term in the sequence 5, 9, 13, 17, ...

2. Find the 101st term of the arithmetic sequence 2, 5, 8, 11, ...

3. In the arithmetic sequence -7, -4, -1, 2, ..., what term is 44?

4. The 2nd term of an arithmetic sequence is 8 and the 4th term is 18. Solve for the common difference, the 1st and the 3rd terms of the sequence.

Answers:

1. an = a1 + (n 1)d

a14 = 5 + (14 1)2

= 5 + 13(2)

a14 = 31

2. an = a1 + (n 1)d

a101 = 2 + (101 1)3

= 2 + (102)3

a101 = 308

3. an = a1 + (n 1)d

44 = -7 + (n 1)3

44 = -7 + 3n 3

44 = -10 + 3n

3n = 54

n = 18

Therefore, 44 is the 18th term in the sequence.

4. Given: a2 = 8; a4 = 18; d = ?; a1 = ? a3 = ?

Solution: There are 3 terms from a2 to a4. Assume a2 as a1 and a4 as a3 to find the common difference, d.

an = a1 + (n 1)d

a3 = a1 + (n 1)d

18 = 8 + (3 1)d

18 = 8 + 2d

2d = 10

d = 5

Using d = 5, therefore the 1st term is 3 and 3rd

term is 15.






Activity 4: Discovered PatternInstructions:

1. Reorganize the class into groups of 4 or as desired.

2. Distribute to each group the activity sheets.

3. Set a time allotment for them to complete the activity.

4. Instruct them to show their solutions on a manila paper for comparison.

5. Check their answers using the answer key on page 28.

This time, let the students to do the challenge:

Ask volunteers to show and discuss their solutions on the board.

After which, present the information below in relation to the given challenge.

Again, without paper and pencil, this was how he reasoned out:

And since there are 50 pairs, each having a sum of 101, Gauss derived the correct sum of 5,050. Discovering the pattern will lead to the sum of the arithmetic series.

What is a series?


Find the sum of the first 100 positive integers, 1 + 2 + 3 + ... + 98 + 99 + 100, without paper and pencil.

Find the sum of the first 100 positive integers, 1 + 2 + 3 + ... + 98 + 99 + 100, without paper and pencil.





Now, ask the class to give the sum of the following series as fast as they can. You may ask volunteers to solve on the board while others are on their seats.

1. 1 + 3 + 5 + 7 + ... + 49.

2. 2 + 4 + 6 + 8 + ... + 60.

3. 5 + 10 + 15 + 20 + ... + 45.

Let us recall the presentation of Karl Gauss in order to obtain the sum of the first 100 positive integers.

Let S be the sum of the arithmetic series; n is the number of terms; a1 is the first term; and d is the common difference. Therefore, we can express the arithmetic series 1 + 2 + 3 + ... + 98 + 99 + 100, as

Sn = 1 + 2 + 3 + 4 + ... + 97 + 98 + 99 + 100

Let us write the series in reverse order,

Sn = 100 + 99 + 98 + 97 + ... + 4 + 3 + 2 + 1

Now, let's add the two series,

2Sn = 101 x 100 = 10,100

Sn=10,100

2=5,050

In symbols, the arithmetic series is,

Sn = a1 + (a1 + d) + (a1 + 2d) + (a1 + 3d) + ... + [a1 + (n 1)d].

In reverse order,

Sn = an + (an - d) + (an - 2d) + (an - 3d) + ... + [a1 - (n 1)d].

Add the two series,

Sn = a1 + (a1 + d) + (a1 + 2d) + (a1 + 3d) + ... + [a1 + (n 1)d]

Sn = an + (an - d) + (an - 2d) + (an - 3d) + ... + [a1 - (n 1)d]

2Sn = (a1 + an) + (a1 + an) + (a1 + an) + . . . + (a1 + an)






Simplify,

2Sn = n (a1 + an)

Sn =n2

(a1 + an)

Since an = a1 + (n 1)d, the formula becomes

Sn =n2

(a1 + a1 + (n 1)d)

Sn =n2

[2a1 + (n 1)d]

Illustrative Example:

Find the sum of the consecutive integers from 1 to 50.

Given: n = 50; a1 = 1; d = 1

Sn =n2

[2a1 + (n 1)d]

=502

[2(1) + (50 1)(1)]

= 25 (2 + 49)

= 25 (51)

Sn = 1,275

When three terms form an arithmetic sequence, the middle number is called the arithmetic mean between the other two. In the sequence for example, 3, 6, 9, 12, ..., 9 is the arithmetic mean between 4 and 8; 6 and 9 are the arithmetic means between 3 and 12.

If a, m, and b is an arithmetic sequence, then m is the arithmetic mean, and m=ab

2.

Solution:






m a = m b

2m = a + b

m= ab2

The arithmetic mean or the mean between two numbers is sometimes called the average of two numbers. It is also possible to insert any number of terms between any two given terms such that the whole set of numbers will form an arithmetic sequence.

Illustrative Examples:

1. Find the arithmetic mean between 12 and 68.

Solution:

12682

=802

= 40

2. Insert three arithmetic means between 4 and 16.

Given: a1 = 4; a5 = 16; n = 5; d = ?

Solution:

an = a1 + (n 1)d

a5 = 4 + (5 1)d

16 = 4 + 4d

4d = 12

d = 3

The sequence is 4, 7, 10, 13, 16. Therefore, the three arithmetic means are 7, 10, and 13.

Activity 5: Summed-up PatternInstructions:

1. Let the same groups of students work this activity.

2. Distribute to them the materials.

3. Set a time allotment for them to accomplish the task.

4. Let them post their outputs for comparison and discussion.

5. Check their outputs using the answer key on page 30.

Formative AssessmentRoam around and check if the students are doing the given tasks correctly and to ensure that everyone in each group has contributed his/her ideas from the different activities.

RoundupThe students should have demonstrated their knowledge and skills related to arithmetic sequence and applied these in solving problems.






4. Check for Understanding of the Topic or SkillThis stage is for teachers to find out how much students have understood before they apply it to other learning experiences.

Background or purposeThe activity suggested in this stage aims to ensure and check how far the students have gained knowledge on arithmetic sequence. They shall solve problems involving arithmetic sequence designated in the different learning stations.

StrategyROTATING LEARNING STATIONS. These are stations which contain setups designed to investigate concepts or perform activities that would allow students to understand a concept. They are installed in strategic places in the classroom where group of students go from one station to another in a round robin manner and do the task indicated in each learning station.

Materials

cut-outs of the following:

Learning Stations 1 to 5

problems on arithmetic sequence, means and series

cartolina

manila paper

graphing paper

pentel pen

masking tape

pencil

Activity 6: Displayed PatternPreparatory Activity:

Prepare the necessary materials prior to the conduct of this activity. Refer to Teacher Resource Sheet 2 on page 31.

Identify five (5) learning stations inside the classroom and paste the Learning Stations 1 to 5 cut-outs and the problems that belong to each station.

Instructions for the Activity:

1. Organize the class into 5 groups.

2. Assign one group in each station to solve the problems with time limit.

3. Let them proceed to the next station in a clockwise direction after giving the signal.

4. The activity will end when they have completed the tasks.

5. Finally, ask them to consolidate their outputs on a manila paper for comparison and discussion.

Formative AssessmentCheck each group's outputs.

RoundupThe students should have solved problems involving arithmetic sequence designated in the different learning stations.






5. Practice and ApplicationIn this stage, students consolidate their learning through independent or guided practice and transfer their learning to new or different situations.

Background or purposeIn this stage, the students will explore and solve some real-life problems involving arithmetic sequence.

StrategyPROBLEM SOLVING. Teaching students how to effectively solve problems will provide them with useful lifelong skills. Problem solving models, such as working mathematically model, break problem solving into a step by step process:

CLARIFY What is the problem asking you to do or find out? What is given in the problem?

CHOOSE What tools would be effective for solving the problem?

USE Use the tools to gain an answer to the problem.

INTERPRET Is this answer reasonable? Can you check it using another method?

Materials

Teacher Resource Sheet 3 on pages 32

manila paper

pentel pen

masking tape

Activity 7: Applied PatternInstructions:

1. Organize the class into eight (8) groups.

2. Assign in every group a leader to facilitate the given tasks.

3. Prepare two sets of each problem under Teacher Resource Sheet 2 in order to have two groups solving a common task.

4. Distribute the needed materials and let them answer the task at a given time.

5. Instruct the two groups with the same task to compare their answers and reach a consensus.

6. Then, let them finalize their answers on a manila paper for presentation.

Formative AssessmentCheck the outputs of the students. Refer to page 33 for the answer key.

RoundupThe students should have explored and solved some real-life problems involving arithmetic sequence.






6. ClosureThis stage brings the series of lessons to a formal conclusion. Teachers may refocus the objectives and summarize the learning gained. Teachers can also foreshadow the next set of learning experiences and make the relevant links.

Background or purposeIn this stage, the students shall list different series of whole numbers to prove/disprove a given assumption.

StrategyMATHEMATICAL INVESTIGATION. An inquiry into a mathematical situation, the topic of which may arise from real life, or from a mathematically designed problem. Students are required to apply familiar skills and concepts to the unfamiliar situation of the investigation and communicate their findings in a report.

Materials


manila paper

pentel pen

masking tape

Activity 8: Unfolded PatternInstructions:

1. Let the same groups of students work on the activity at a given time allotment.

2. Distribute to each group the needed materials.

3. Let them post their outputs for comparison and discussion.

Formative AssessmentRoam around to ensure the participation of the students as they perform the task.

RoundupThe students should have listed different series of whole numbers to prove/disprove a given assumption.

Teacher Evaluation(To be completed by the teacher using this Teachers Guide)

The ways I will evaluate the success of my teaching this unit are:

1.

2.

3.






STUDENT ACTIVITY 1Unique Pattern

Objectives: Determine and draw correctly the fourth figure of the sequences.

Directions: Each of the following items below shows a sequence of figures. Determine and draw correctly the fourth figure in the box given the first three terms.

1.

2.

3.

4.

5.






6.

7.

8.

9.

10.






STUDENT ACTIVITY 1Unique Pattern

Possible Solutions

1. 6.

2. 7.

3. 8.

4. 9.

5. 10.






TEACHER RESOURCE SHEET 1Hidden Pattern

Directions: Reproduce 5 sets of the following figure strips and cut.






STUDENT ACTIVITY 2Hidden Pattern

Objectives:


Derive, by pattern-searching, a mathematical expression (rule) for generating a sequence.

First three illustrations:

Questions:

1. Before you discovered the pattern of the given series of figures, do you think the succeeding one can easily be determined by just looking at the first two or three figures? Why or why not?

2. What is the sequence you have discovered?

3. What have you observed when you subtract a number from its succeeding term in the given order?

4. What are the five numbers before 2?

5. Now, list your new series of numbers. What is its 9th term?

6. What is the sum of all the numbers you listed?






STUDENT ACTIVITY 3Reflected Pattern

Procedure:

1. Form squares out of matchsticks as illustrated.

2. Continue the process and record the results in the table below.

number of squares (n) 1 2 3 4 5 6 7 8 9 10 n

number of matchsticks (m) 4 7 10

3. Plot the set of ordered pairs on a graphing paper as described in the table. Describe the graph of the given sequence.

Answer the following:

1. How does the number of matchsticks relate to the number of squares?

______________________________________________________________________________________________________________________________________________________________________________________________

2. Express their relationship in a mathematical sentence.

______________________________________________________________________________________________________________________________________________________________________________________________

3. Determine the total number of matchsticks if there will be 100 squares.

______________________________________________________________________________________________________________________________________________________________________________________________






STUDENT ACTIVITY 4Discovered Pattern

Objective: Solve problems on arithmetic sequence.

1. Find the 20th term of the arithmetic sequence 17, 13, 9, ...

6. If a1 = 5, a7 = 17 and n = 7, find the common difference.

2. In the arithmetic sequence 8, 5, 2, -1, ..., which term is -25?

7. Determine the 2nd term of the arithmetic sequence in which the 4th term is -2 and the common difference is 6.

3. The 3rd term of an arithmetic sequence is 0 and the 5th term is 6. Solve for the common difference.

8. The 16th term of an arithmetic sequence is 56. If the common difference is 3, find the first term.

4. Find the 31st term of the sequence -3, 2, 7, 12, ...

9. What is the 1001st term of the sequence 2, 4, 6, 8, ...?

5. If a1 = 5, an = 241, and d = 4, find the value of n.






STUDENT ACTIVITY 4Discovered Pattern

Answer Key

1. an = a1 + (n 1)d

a20 = 17 + (20 1)(-4)

= 17 + (19)(-4)

= 17 76

= -59 (R)

2. an = a1 + (n 1)d

-25 = 8 + (n 1)(-3)

-25 = 8 3n + 3

-25 = 11 3n

-3n = -36

n = 12 (O)

3. an = a1 + (n 1)d

Given: a3 or a1 = 0; a5 or an = 6

6 = 0 + (3 1)d

6 = 0 + 2d

d = 3 (T)

4. an = a1 + (n 1)d

a31 = -3 + (31 1)5

= -3 + (30)(5)

= -3 + 150

= 147 (L)

5. an = a1 + (n 1)d

241 = 5 + (n 1)4

241 = 5 + 4n 4

241 = 1 + 4n

240 = 4n

n = 60 (U)

6. an = a1 + (n 1)d

17 = 5 + (7 1)d

17 = 5 + 6d

12 = 6d

d = 2 (G)

7. an = a1 + (n 1)d

Given: let a2 = a1; a4 = a3

-2 = a1 + (3 1)6

-2 = a1 + 2(6)

-2 = a1 + 12

a1 = -14 (A)

8. an = a1 + (n 1)d

56 = a1 + (n 1)d

56 = a1 + (16 1)(3)

56 = a1 + 45

a1 = 11 (Y)

9. an = a1 + (n 1)d

a1001 = 2 + (1001 1)(2)

= 2 + 2000

= 2,002 (E)

The hidden message:

LETTER Y O U A R E A L L G R E A T

ITEM # 8 2 5 7 1 9 7 4 4 6 1 9 7 3






STUDENT ACTIVITY 5Summed-up Pattern

Objectives:

Find the sum of an arithmetic sequence.

Solve problems involving arithmetic means.

Word/s

Item Numbers 4 1 6 3 2 5






STUDENT ACTIVITY 5Summed-up Pattern

Answer Key

1. an = a1 + (n 1)d

Given: a1 = -9; a5 = 7; n = 5; d = ?

a5 = -9 + (5 1)d

7 = -9 + 4d

16 = 4d

d = 4

The sequence is -9, -5, -1, 3, 7. Therefore, the three arithmetic means

are , -1, and 3. [TIME IS A]

4. Sn =n2

[2a1 + (n 1)d

S10 =102

[2(2) + (10 1)4

= 5 (4 + 36)

= 5 (40)

= [WELL-ARRANGED]

2. 1767

2

=502

=

[WELL-ARRANGED]

5. Sn =n2

[2a1 + (n 1)d

S12 =122

[2(9) + (12 1)(-2)

= 6 (18 22)

= 6 (-4)

= [MIND]

3. Given: a1 = 27; a4 = -3; n = 4; d = ?

an = a1 + (n 1)d

a4 = 27 + (4 1)d

-3 = 27 + 3d

-3 27 = 3d

d = -10

The sequence is 27, 17, 7, -3. Therefore, the two arithmetic means are 17 and

. [OF A]

6. Sn =n2

[2a1 + (n 1)d

S50 =502

[2(1) + (50 1)(2)

= 25 (2 + 98)

= 25 (100)

= [MARK]

Word/s Well-arranged

time is a mark of a well-arranged

mind.

Item Numbers 4 1 6 3 2 5


-5

25

7

200

-24

2,500





TEACHER RESOURCE SHEET 2Displayed Pattern

Directions:

1. Prepare the following cut-outs using cartolina.

2. Write the following problems for every learning station on a manila paper.

Learning Station 1:

1. Write the first six terms of the arithmetic sequence when a1 = 15 and d = 3.

2. Find the sum of the first twelve terms of the arithmetic series 2 + 5 + 8 + ...

3. Find the arithmetic mean between 9 and 49.

Learning Station 2:

1. Write the first three terms of the sequence if the 4th term is 11 and d = -3.

2. Find the 75th term of the sequence -13, -8, -3, 2, ...

Learning Station 3:

Give the next 5 terms of the sequence -4, -2, 0, 2, ..., and draw its graph.

Learning Station 4:

In the arithmetic sequence -5, -2, 1, 4, ..., what term is 58?

Learning Station 5:

A blacksmith offered to shoe a horse on the following conditions: 1 for the first nail, 3 for the second nail, 5 for the third nail, and so on. If there are 7 nails in each shoe, how much would it cost to shoe a horse?






TEACHER RESOURCE SHEET 3Applied Pattern

Directions: Reproduce each of the problems below in two sets.


1. Cathy saved 10 pesos on the first day of the year, 13 pesos on the second day, 16 pesos on the third day, and so on, up to the end of the year. How much did she saved on the 365th day?





ANSWER KEYApplied Pattern

1. Given: a1 = 10 pesos; d = 3;

a365 = ?

an = a1 + (n 1)d

a365 = 10 + (365 1)3

= 10 + (364)3

= 10 + 1092

= 1,102.00

3. Given: 1999 is a1 = 12 fruits; d = 3

2012 is a14 = ?

It will take 14 years from 1999 to 2012.

an = a1 + (n 1)d

a13 = 12 + (14 1)(3)

= 12 + 13(3)

= 12 + 39

= 51 fruits

2. Given: a1 = 60; d = -6

a8 = ?; a10 = ?

an = a1 + (n 1)d

a8 = 60 + (8 1)(-6)

= 60 42

= 18 blocks

a10 = 60 + (10 1)(-6)

= 60 54

= 6 blocks

The 10th row is indeed the top row.

4. a1 = 25; d = 1; n = 30

Sn =n2 [2a1 + (n 1)d]

=302

[2(25) + (30 1)1]

= 15 (50 + 29)

= 15 (79)

= 1,185 seats






STUDENT ACTIVITY 8Unfolded Pattern

Objective: List different series of whole numbers to prove/disprove the given assumption.

Reference: Mathematical Adventures for Teachers and Students by Wally Green

Examples:

9 = 4 + 5 and also 2 + 3 + 4

15 = 7 + 8 and 4 + 5 + 6 and also 1 + 2 + 3 + 4 + 5






For the Teacher: Translate the information in this Learning Guide into the following matrix to help you prepare your lesson plans.

Stage 1. Activating Prior Learning

2. Setting the Context

3. Learning Activity Sequence

4. Check for Understanding

5. Practice and Application

6. Closure

Strategies

Activities from the Learning Guide

Extra activities you may wish to include

Materials and planning needed

Estimated time for this Stage

Total time for the Learning Guide Total number of lessons needed for this Learning Guide


Mathematics Arithmetic Sequences

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