10

Mathematics
Quarter 1 – Module 1:
Generating Patterns
Lesson 1 Generating Patterns

WHAT’S IN
When you were in grade 8, you learned about concepts related to generating
patterns like Inductive Reasoning. The knowledge and skills you acquired are
very important for you to understand how to generate patterns and
sequences. Hence, let us review inductive reasoning and perform the
activities that follow.

Inductive Reasoning is the process of observing data, recognizing patterns,

and making generalizations (conjecture) from observations. A conjecture is a
conclusion made from observing data or an educated guess based on patterns.

Activity 1. Complete Me!

Direction: Make a conjecture about each pattern. Then use your conjecture
to draw or write the next term in the pattern.
A) Complete the puzzle below by providing the needed term/s in each pattern.
ACROSS
3) 1, 8, 27, 64, 125, ___
5) 4, 20, 100, 500, ___
7) 1, 1, 2, 3, 5, 8, ___
8) J, F, M, A, M, J, J, A, __, __, __,
__
DOWN
1) 2, 4, 6, 8 10, ___
2) 128, 64, 32, ___
4) 1, 10, 100, 1000, ___
5) 3, 9, 27, 81, ___
6) 1A, 2B, 3C, ____
9) O, T, T, F, F, S, S, E, ___, ___
 B) Draw the next object in each picture pattern.

What’s New
At this point of the module, you are about to learn the Introduction to
Sequences and Patterns. To understand better how to generate patterns, you
will have to perform the simple activity below.

Activity 2. Let’s Discover!

Direction: Read, understand, and perform the given instructions below
then answer the questions asked. Materials: string, pair of scissors

1) Prepare five (5) strings with equal lengths.

2) Cut the first string once. (a)How many pieces are there? _____
Cut the second string twice. (b) How many pieces are there?_____
Cut the third string thrice. (c) How many pieces are there?_____
Cut the fourth string four times. (d) How many pieces are there?_____
Cut the fifth string five times. (e) How many pieces are there?_____

3) Based from your answers, complete the table below.

Number of cuts (x) 1 2 3 4 5
Number of pieces (y)

4) Without cutting a string 6 times, how many pieces are there? _____

5) Have you seen a pattern? If yes, describe the pattern and state your
conjecture. Use a formula or equation in your conjecture, where y is the
number of pieces and x is the number of cuts.

6) Using your conjecture, how many pieces of strings can be made from
(a)12 cuts? (b) 24 cuts? (c) 35 cuts? and (d) 42 cuts? Show your solutions.
Were you able to complete the task? If yes, you may proceed to the next page.
If no, take time to finish for you to better understand the next discussions.

What Is It
How did you find activity 2? Have you given idea on how to generate a
pattern? Let us process your answers.
1. Based from the task, the complete solution is shown in the table below:
Number of cuts (x) 1 2 3 4 5 6
Number of pieces (y) 2 3 4 5 6 7
2. From the table, notice that the number of pieces (y) of strings is one more
than the number of cuts (x). Thus, we can state our conjecture as, “The
number of pieces (y) when a string is cut x times can be computed using
the formula y = x + 1.”

3. Using the formula y = x + 1, we can now solve the number of pieces of

strings that can be made from 12 cuts? 24 cuts? 35 cuts? and 42 cuts?
a) 12 cuts, 12 y = 12 + 1 = 13
b) 24 cuts, 24 y = 24 + 1 = 25
c) 35 cuts, 35 y = 35 + 1 = 36
d) 42 cuts, 42 y = 42 + 1 = 43
Were you able to get the same answers? If yes, very good! If no, I hope you
were able to understand the discussions above. Based from the given activity,
the number of pieces, 1 , when a string is cut times represents a sequence.
Thus, the values of which are 2, 3, 4, 5, 6, ,... is an example of a sequence.

The word sequence means an order in which one thing follows another in
succession. A sequence is an ordered list. For another example, if we write x,
2x 2, 3x3, 4x 4, 5x5, ? , what would the next term in the sequence be—the one
where the question mark now stands? The answer is 6x6.

Definition

A sequence is a set of objects which is listed in a specific order, one after

another. Each member or element in the sequence is called term. The terms
in a sequence can be written a1, a2, a3, a4, ……, an, …. which means a1 is
the first term, a2 is the second term, a3 is the third term, …., an is the nth
term and so on.
Sequences are classified as finite and infinite. A finite sequence contains a
limited number of terms. This means it has an end or last term. Consider the
examples below.
a) Days of the week: { Sunday, Monday, Tuesday,…., Saturday}
b) First 10 positive perfect squares: 1, 4, 9, 16, 25,36, 49, 64, 81, 100}
On the other hand, an infinite sequence contains a countless number of
terms. The number of terms of the sequence continues without stopping or it
has no end term. The ellipsis (…) at the end of the following examples shows
that the sequences are infinite. Consider the examples below.
a) Counting numbers: {1, 2, 3, 4, 5,….}
b) Multiples of 5: { 5, 10, 15, 20, 25,…..}
Sometimes a pattern in the sequence can be obtained and the sequence can
be written using a general term. In the previous example x, 2x2, 3x3, 4x4,5x5,
6x6,..., each term has the same exponent and coefficient. We can write this
sequence as an = nxn , n =1, 2, 3, 4, 5, 6,..., where is called the general or
nth term.
A. Finding several terms of a sequence, given the general term:
Example 1.

Find the first four terms of the sequence an = 2n − 1.

Solution: To find the first term, let n = 1

an =2n − 1 use the given general term

a1 = 2 −1 substitute by 1

a1 = 2 − 1 perform the operations

a1=1 simplify
Repeat the same process for the second to the fourth terms.
Find the second term, a2 a2 = 2(2) -1 = 4 -1 = 3
Find the third term, a3 a3 = 2(3) –1 = 6 -1 = 5
Find the fourth term, a4 a4 = 2(4) –1 = 8 -1 = 7
Therefore, the first four terms of the sequence are 1, 3, 5, 7.

Example 2.

Find the 5th to the 8th terms of the sequence bn = (-1)n / n+1.

Solution: Find the 5th term, let n = 5

bn = (-1)n / n+1 use the given general term

 b5 = (-1)5 / 5+1 substitute n by 5

b5 = - 1/6 simplify (− 1 raised to an odd number power is always negative )

Repeat the same process for the 6th to the 8th terms.

Find the 6th term, n =6 n6 =

Find the 7th term, n=7 n7 =

Find the 8th term, n =8 n8 =

Therefore, the 5th to the 8th terms of the sequence are -1/6, 1/7, -1/8, 1/9.

B. Finding the general term, given several terms of the sequence:

Example 3.

Write the general term of the sequence 5, 12, 19, 26, 33,...

Solution: Notice that each term is 7 more than the previous term. We can
search the pattern using a tabular form.
Term Given Pattern
1 5 5 5 + 7(0)
2 12 5+7 5 + 7(1)
3 19 5+7+7 5 + 7(2)
4 26 5+7+7+7 5 + 7(3)
5 33 5+7+7+7+7 5 + 7(4)
n an 5 + 7 + 7 + 7 + 7 +…+ 7 5 + 7(n – 1)
In the pattern, the number of times that 7 is added to 5 is one
less than the nth term (n – 1). Thus,
an = 5 5+7(n – 1) equate an and 5 + 7(n−1)
+
an = 5 5+ 7n -7 apply distributive property of multiplication

an = 7n - 2 combine similar terms

Therefore, the nth term of the sequence is an = 7n -2 , where 1, 2, 3, 4, 5,...

Example 4.

Write the general term of the sequence 2, 4, 8, 16, 32, ...

Solution: Notice that each term is 2 times the previous term. We can search
the pattern using a tabular form.
 Term Given Pattern

1 2 2 21
2 4 2(2) 22
3 8 2(2)(2) 23
4 16 2(2)(2)(2) 24
5 32 2(2)(2)(2)(2) 25
n an 2(2)(2)(2)(2)…(2) 2n
Therefore, the nth term of the sequence is an = 2n, where n = 1, 2, 3, 4, 5,...

Example 5.

Find the general term of the sequence 1, , , , ,...

Solution: , , , , ,... write 1 as

1 1 1 1 1
1 2 2 2
1 2 3 4 52
Notice each denominator is an integer squared

Therefore, the nth term of the sequence is an = 1/ n2, where n =1, 2, 3, 4, 5,...

What’s More
Now, it’s your turn to apply the concepts on sequences and patterns to find
the specified terms of a sequence when given its general term and vice versa.
Activity 3. Your Turn!

Direction: Answer what is asked in each set of exercises on a separate sheet

of paper.

• In Exercises 1 – 4, write the first four terms of each sequence. Assume

starts at 1.

1) an = n 3) an = (-1)n+1n2
2) an = n / n+1 4) an = n(n+1) / 2
• In Exercises 5 – 8, find the indicated term of each sequence given.

5) an = ( ½ )n a9 = ? 7) an = (-1)n+1(n-1)(n+2) /n a7 =?

6) an = (n+1)2 / n-9 a14 =? 8) an = (n/9 -12)n a99 = ?

∙ In Exercises 9 – 12, write an expression for the nth term of the given
sequence. Assume n starts at 1.

9) 2, 4, 6, 8, 10,… 10) -1, 1, -1, 1, ….

11) 3, 9, 27. 81,… 12) 1 / 2*1, 1/ 3*2, 1/ 4*3, 1/ 5*4, 1/ 6*5,…

What I Have Learned

Great! You have reached this part of the module. To ensure your full
understanding on the concepts related to generating patterns, it’s important
that you are able to complete each statement below.

A sequence is ________________________________________________________
A term is ____________________________________________________________
A finite sequence is ________________ while infinite sequence is
_______________
To find the specified term/s of a sequence when given the general term,
___________
To write the general term of a sequence when given some terms,
________________

What I Can Do
Generating patterns is a vital concept in performing any mathematical
investigation. Similar to the previous activity on the number of pieces when
a string is cut x times, a sequence is formed when a repeated process
following a certain rule is employed. To perform a simple mathematical
investigation, perform the task below.
Activity 4. Let’s Investigate!

Direction: Given the figures below, perform a mathematical investigation by

following the given steps.

Figure 1 Figure 2 Figure 3 Figure 4

Step 1. To investigate, you are going to make your own problem to solve.
Based from the figures above, what do you think could be a probable
problem to investigate?
Step 2. Identify your variables (dependent and independent variables). For
example, number of pieces of strings, , and number of cuts . Then list
your data.
Step 3. Present your data in a tabular form.
Term Sequence Pattern
1
2
3

Step 4. State your conjecture using a general or nth term.

Step 5. Test your conjecture. Using your derived formula or general term, find
the first four terms of your sequence if it matches.
 Assessment
Let us determine how much you have learned from this module. Read and
understand each item, then choose the letter of your answer and write it on
your answer sheet.

1. Which of the following shows a pattern?

A) 3, 2, 3, 2, 3, 2 C) A, G, M, T, 0, 9

B) B) 5, 7, 2, T, 6, Y D) smooth, long, wall, sing

2. Which of the following defines infinite sequence?

A) days of the week C) every other day

B) teenage life D) first Fridays of July 2020

3. What are the next three terms of the sequence 1, 11, 22, 34, ....?
A) 46, 6 , 75 B) 4 , 61, 76 C) 42, 54, 66 D) 44, 6 , 74

4. What is the 25th term of the sequence an = ?

A) B) − C) D) −

5. What is the 11th term in the sequence − 1, 4, − 9, 16, − 25, ..?

A) 100 B) – 100 C) 121 D) − 121

6. Find the first four terms of the sequence an = 3n + 2..

A) 5, 7, 11, 14 B) 5, 8, 11, 15 C) 5, 8, 11, 14 D) 5, 9, 13, 17

7. Which numerical pattern follows the rule “subtract 2, then multiply

by 3”, when starting with 5?
A) 5, 7, 21, 69 B) 5, 3, 9, 7, 21 C) 5, 3, 6, 4, 12 D) 5, 9, 14, 36

8. What rule will correctly describe the sequence: 2, 6, 12, 20, 30,…?
A) n+1 B) n2 +1 C) 2n +1 D) an = - 1/ 2n

9. Find the nth term of the sequence – ½, ¼, - 1/6, 1/8, - 1/10,…

A) an = -1/ 2n B) an = (-1)2/2 C) an = (-1)2/2n D) an = 1/ 2n

10. Which is the next ordered pair in the pattern (2, 1), (4, 4), (6, 9) ?
A) (8, 12) B) (7,16) C) (8, 16) D) (7, 12)

11. What is the 8th term in the sequence 9, 4, -1, -6, -11, …?
A) − 21 B) − 26 C) − 31 D) – 36
−1 1 −1 2
12. In the sequence, an =(-1)n+1(n-1)(n+2) /n , what is a10?

A) B) − C) D) −

13. Which of the following patterns shows finite sequence?

A) 6, 12, 1 , 24, 30 ,... C) First 20 whole numbers
B) multiples of 6 D) 100, 50 , 25, 12.5,...

14. Find the general term of the sequence 3, 9, 27 , 81,...

A) 3n B) n3 C) 3n D) n+3

15. Write the first four terms of the sequence an = n 2 − 1.

A) 0, 3, 8, 15 B) 1, 3, 5, 7 C) 1, 5, 10 , 16 D) 0, 2, 7, 12

Additional Activities
Awesome! Before we end this module, let me introduce a puzzle game called
Tower of Hanoi. Are you familiar with this game? If not, allow me to introduce
it to you.

In the Tower of Hanoi puzzle

a player attempts to move a large pile
of disks, known as the Tower, from
the leftmost peg to the rightmost on
the puzzle board. The rules of the
puzzle state that the player can only
move one disk per turn and can
never place a larger disk onto a
smaller one at any time.
If you are interested to play the puzzle, you can search it on the internet. For
the meantime, I just want to use the puzzle game for the purpose of applying
sequences and generating patterns.
Situation: In playing the game, you can choose the number of disks of your
tower and play with the least possible moves. The least number of
moves when playing the puzzle with respect to the number of disks
are as follows:
Number of Number of
Number of Disks Number of Disks
Moves Moves
1 1 5 31
2 3 6 63
3 7 7 127
4 15 9 255
Task: Based on the above data, find the general term in finding the number
of moves with respect to the number of disks in playing the tower of
Hanoi.
Module 1- Quarter 1, Week 1

Answer Sheet

Name:_________________________________ Yr/Sec: _______________________

Activity 1. Complete Me!

A. B. 1. 4.

2. 5.

3. 6.

Activity 2. Let’s Discover! Activity 3. Your Turn!

1. 1.
2. a) 2.
b) 3.
c) 4.
d) 5.
e) 6.
3. 7.
No. of cuts (x) 1 2 3 4 5 8.
9.
No. of pieces (y)
10.
4. 11.
5. 12.
6. a)
b)
c)
d)
Activity 4. Let’s Investigate!

Term Sequence Pattern

1
2

3
4

 State your conjecture using a general or nth term.

 Test your conjecture. Using your derived formula or general term, find the
first four terms of your sequence if it matches.

ASSESSMENT
1. 6. 11.
2. 7. 12.
3. 8. 13.
4. 9. 14.
5. 10. 15.

Additional Activities