math 8 MATATAG Curriculum week 1 quarter 1.pptx

1. Week 8: Day 1
Introduction to sequences

2. What is a sequence?
A mathematical sequence is a series of numbers connected by a rule.
This rule defines their pattern
3, 8, 13, 18, 23 +5 every time
2, 4, 8, 16, 32 x2 every time
11, 7, 3, -1, -5 -4 every time
Each part of a sequence is
called a ‘term’
How we move from one term to the next is
called the term to term rule.

3. Types of sequence
There are two key types of sequences.
Arithmetic sequences
These have a term to term rule which is either an addition or a subtraction.
Geometric sequences
These have a term to term rule which is either an multiplication or a division.

4. Finite Sequence contains a limited number of terms.
Infinite Sequence contains countless number of terms.

5. Finding specific terms
What is the 7th
term in this sequence?
2, 5, 8, 11…
+3 +3 +3 +3
14
+3
17
+3
20
What is the 70th
term in this sequence?
We need a better method than just moving along the sequence!

6. Fortunately all sequences also have a position to term rule
2, 5, 8, 11, 14, 17, 20 3n – 1
The position we are looking for is ‘n’
So the 7th
term is… 3 x 7 - 1 = 20
And the 70th
term is… 3 x 70 - 1 = 209
We often call this the nth
term rule.
Finding specific terms

7. Formulating Rules for sequence

8. Part I: Study the following sequences below. Write
the rule describing each sequence.
Sequence nth Term Rule
1.) 3,6,9,12...
2.) 1,4,9,16...
3.) y,...
4.) 2,5,8,11...
5.) 2,4,6,8….
3n
𝒏𝟐
𝒚 𝒏
3n-1
2n

9. Sequence The next three term
6. 5, 7 , 9, 11
7. 15, 18, 21, 24
8. 3, 8, 18, 38
9. 7, 10, 13, 16,
10. 18, 27, 36, 45,
13, 15, 17,
27, 30, 33
78, 158, 318
19, 25
22,
54, 63, 72

10. SEQUENCE
Is a set of numbers written in
order by the application of a
definite rule.

11. SEQUENCE
Example:
3, 5, 7, 9…
+2 +2 +2
11
+2

12. SEQUENCE
3, 5, 7, 9…
TERMS
1st
term 2nd
term 3rd
term 4th
term

13. Examples of Sequence:
3, 6, 9, 12…
1, 4, 9, 16…
2, 5, 8, 11 …

14. 3, 6, 9, 12…
How to formulate the rule?
What i s the nth Term Rule?
What are the Next Three Terms?

15. 3, 6, 9, 12…
3, 6, 9, 12…
How to formulate the Rule?
1st
term 2nd
term 3rd
term 4th
term
3 x 1 3 x 2 3 x 3 3 x 4
What is the nth Term Rule? 3n

16. What are the next three terms?
3, 6, 9, 12… nth Term Rule = 3n
(3 x n)
3, 6, 9, 12, ____, _____, _____,
1st
term
2nd
term
3rd
term
4th
term
5th
term
6th
term
7th
term
= 3n
= 3 x 5
= 15
15
= 3n
= 3 x 6
= 18
18
= 3n
= 3 x 7
= 21
21

17. What is 20th
term in the sequence?
3, 6, 9, 12, 15, 18, 21… _______________
nth Term Rule = 3n
= 3n
= 3 x 20
= 60

18. 2, 5, 8, 11 …
How to formulate the rule?
What i s the nth Term Rule?
What are the Next Three Terms?

19. 2, 5, 8, 11 …
How to formulate the Rule?
2, 5, 8, 11
5 – 2 = 3 8 – 5 = 3 11 – 8 = 3

20. Guess: 3n + 1
2, 5, 8, 11 …
2, 5, 8, 11
5 – 2 = 3 8 – 5 = 3 11 – 8 = 3
Check: 3n + 1
= 3(1) + 1
= 3 + 1
= 4
What is the nth Term Rule?

21. Guess: 3n- 2
2, 5, 8, 11 …
2, 5, 8, 11
5 – 2 = 3 8 – 5 = 3 11 – 8 = 3
Check: 3n- 2
= 3(1)- 2
= 3- 2
= 1
What is the nth Term Rule?

22. Guess: 3n- 1
2, 5, 8, 11 …
2, 5, 8, 11
5 – 2 = 3 8 – 5 = 3 11 – 8 = 3
Check: 3n- 1
= 3(1)- 1
= 3- 1
= 2
What is the nth Term Rule? 3n- 1

23. What are the next three terms?
2, 5, 8, 11 …
nth Term Rule = 3n-1
2, 5, 8, 11, ____, ____, _____
= 3n - 1
= 3 (5) - 1
= 15 - 1
= 14
14
= 3n - 1
= 3 (6) - 1
= 18 - 1
= 17
17
= 3n - 1
= 3 (7) - 1
= 21 - 1
= 20
20

24. What is the 50th
term in sequence?
2, 5, 8, 11 …
nth Term Rule = 3n-1
= 3n - 1
= 3 (50) - 1
= 150 - 1
= 149

25. How to formulate the rule?
1, 4, 9, 16…
1, 4, 9, 16…
4-1
+3 +5 +7
9-4 16-9
+9
25

26. How to formulate the rule?
1, 4, 9, 16…
1, 4, 9, 16…
1 X 1 2 X 2 3 X 3 4 X 4
What is the nth Term Rule? 𝒏𝟐

27. What are the next three terms
in the sequence?
1, 4, 9, 16… nth term rule: 𝒏𝟐
1, 4, 9, 16, ___, ____, ____,
𝒏𝟐
𝟓𝟐
25
25
𝒏𝟐
𝟔𝟐
36
𝒏𝟐
𝟕𝟐
49
36 49

28. 1, 4, 9, 16…
What is the 15th
term in sequence?
nth term rule: 𝒏𝟐
𝒏𝟐
𝟏𝟓𝟐
225
1, 4, 9, 16… 225

30. Direction: Write the rule to find the nth term of the
given sequence. Supply the next three terms after.
Sequence nth Term
Rule
Next Three
Terms
1. 5, 9, 13, 17…
2. 3, 8, 13, 18…
3. 6, 12, 18, 24, 30…
4 6, 13, 20, 27…
5. 7,11,15,19, …

31. GEOMETRIC
SEQUENCE

32. OBJECTIVES
After going through this module, the learner should be
able to:
a. determine a geometric sequence;
b. identify the common ratio of a geometric sequence;
c. find the missing term of a geometric sequence; and,
d. determine whether a sequence is geometric or arithmetic.
e. insert geometric means in a sequence
f. solve the sum of a geometric sequence.

33. WHAT IS A GEOMETRIC SEQUENCE?
A geometric sequence is a
sequence obtained by
multiplying a common ratio to
the preceding terms in order to
obtain the succeeding terms.

34. HOW TO DETERMINE THE COMMON RATIO?
The common ratio is obtained by
dividing a term by the term preceding it.
, , , …

35. EXAMPLES
1, 2, 4, 8, 16, … r = 2
3, -6, 12,-24, … r = -2
3, 1, , , … r =

36. ACTIVITY 1: I’LL TELL YOU WHAT YOU ARE
State whether each of the following sequences is geometric or not.
1. 5, 20, 80, 320, …
2. 7, 5, 3, , …
3. 5, -10, 20, -40, …
4. 1, 0.6, 0.36, 0.216
5. 10/3, 10/6, 10/9, 10/15, …
Geometric
Not Geometric
Geometric
Geometric
Not Geometric

37. Identify which of the following is a geometric sequence. If observed as a
geometric sequence, determine r.
1. 1, 2, 4, 12, 16, … 6. 1, 3, 9, 27, 81, . . .
2. -2, 4, -8, , -32, … 7. 27, 9, 3, 1, 1/3, …
3. 10, 20, 30, 40, 50… 8. 3, 12, 48, 192, 768, . . .
4. 5, 10, 20, 40, 80, … 9. 2, 26, 338, 4 394, …
5. 4, 12, 36, 108, 324 … 10. 5x2
, 5x5
, 5x8
, 5x11
, 5x14
, …
X
r = -2
X
r = 2
r = 3
r = 3
r = 1/3
r = 4
r = 13
r =
ACTIVITY 2: GEOMETRIC BA ‘YAN?

38. Arithmetic
Sequence
Geometric
Sequence
common difference
common ratio
vs
an = a1 + (n-1)d
an = a1rn-1

39. ACTIVITY 3: SHADE THAT BOX
WORKSHEET
Shade the box with
blue if the sequence
inside it is geometric,
but color it red if it
contains an arithmetic
sequence. Leave the
box uncolored if
neither a geometric nor
arithmetic.

40. ACTIVITY 3: SHADE THAT BOX (ANSWER)

41. FINDING NTH
GEOMETRIC SEQUENCE

42. an = a1rn-1
FORMULA
nth term
or last term first term
common ratio
position or order
of the term

43. EXAMPLE
Find the 9th
term of the sequence
1, 3, 9, 27, …
a1 = 1
r = 3
n = 9
an = a1rn-1
a9 = (1)(3)9-1
a9 = (1)(3)8
a9 = (1)(6561)
a9 = 6561

44. EXAMPLE
Find the 10th
term of the sequence
-2, 8, -32, 128, …
a1 = -2
r = -4
n = 10
an = a1rn-1
a10 = (-2)(-4)10-1
a10 = (-2)(-4)9
a10 = (-2)(-262,144)
a10 = 524,288

45. EXAMPLE
What is the 5th
term of the geometric
sequence whose a1 = , and r = 2?
an = a1rn-1
a5 = (2)5-1
a5 = (2)4
a5 = (16)
a5 =

46. TRY THESE!

47. ARITHMETIC
SEQUENCE

48. After going through this module,
you are expected to:
1. define an arithmetic
sequence,
2. illustrate an arithmetic
sequence and
3. finding the nth term of an

49. MIND EXERCISE
DETERMINE THE NEXT THREE
TERMS OF THE SEQUENCES.
1. 5, 12, 19, 26, …
2. 33, 38, 43, 48, …
3. -14, -10, -6, -2, …
33, 40, 47
53, 58, 63
2, 6, 10

50. How did you
determine the
terms in the
sequence?

51. Arithmetic
Sequence
A sequence whose consecutive
terms have a common difference is
an arithmetic sequence.
The sequence a1, a2, a3, …, an is
arithmetic if there is a number d such
that:
a2 – a1 = d, a3 – a2 = d, a4 – a3 =
d.
d is called the common

52. 2, 5, 8, 11,
14, …
Is this an arithmetic
sequence?
3 3 3 3
d =

53. -1, -4, -7, -11, -
14, …
What about this?
-3 -3 -4 -3
There is no common

54. ACTIVITY 1: ARITHMETIC OR NOT?
WHICH OF THE FOLLOWING IS AN ARITHMETIC
SEQUENCE? IF THE SEQUENCE IS ARITHMETIC, DETERMINE
THE COMMON DIFFERENCE, BUT WRITE NA IS NOT AN
ARITHMETIC SEQUENCE.
1. 3, 0, -3, -6, -8,
…
2. -15, -10, -5, 0,
5, …
3. 2, 13, 15, 28,
43, …
4. 17, 23, 29, 35,
NA
d = 5
NA
d = 6
d =

55. ACTIVITY 2: WHAT’S THE DIFFERENCE?
1. 27, 21, 15, 9,
3, …
2. 6, 10, 14, 18,
22
3. 6, 4, 2, 0, -2,
…
4. -20, -13, -6, 1,
8, …
d = -6
d = 4
d = -2
d = 7
d = 5
Determine the common difference of the
following arithmetic sequences:

56. ACTIVITY 3: COMPLETE ME!
THE FOLLOWING SEQUENCES ARE
ARITHMETIC. COMPLETE THEM BY SUPPLYING
THE CORRECT TERM/S.
1. 5, 9, 13, __, 21,
…
2. -14, -7, __, 7, 14,
…
3. __, 10, 22, 34,
46, …
4. 50, __, 28, 17, 6,
…
17
0
-2
39
8.2

57. FINDING THE
NTH TERM
OF AN
ARITHMETIC
SEQUENCE
02

58. What is the 6th
term
of the sequence 4,
11, 18, 25, …?
Answer: 39

59. What is the 75th
term of the
sequence 4, 11, 18,
25, …? HOW???

60. A
an = a1 + (n-
1)d
an = the nth term
a1 = first term
n = number of terms/the position
of the term being solved
d = common difference

61. an = a1 + (n-
1)d
an =
a75
a1 = 4
n =
75
What is the 75th
term of the
sequence 4, 11, 18, 25, …?
a75 = 4 + (75 –
1)(7)
a75 = 4 + (74)
(7)
a75 = 4 + 518

62. an =
a9
a1 = -
1
n = 9
d = 6
What is the 9th
term of the
sequence -1, 5, 11, 17, 23,
…?
an = a1 + (n –
1)d
a9 = -1 + (9 – 1)
(6)
a9 = -1 + (8)(6)
a9 = -1 + 48

63. an =
a11
a1 = -
4
n =
11
What is the 11th
term of
the sequence whose first
term is -4 and common
difference of 3?
an = a1 + (n –
1)d
a11 = -4 + (11 –
1)(3)
a11 = -4 + (10)
(3)

64. an =
a7
a1 =
23
n = 7
d = -5
If a1 = 23, and d = -5, what
is a7?
an = a1 + (n –
1)d
a7 = 23 + (7 –
1)(-5)
a7 = 23+ (6)(-5)
a7 = 23 – 30

65. Challenge Question
If a13 = 112, and d =
-8,
what is the first

66. ACTIVITY 1: What’s
my nth?
Determine the indicated term of
each arithmetic sequence.
1. 2, 8, 14, 20, 26, …, a21
2. -12, -7, -2, 3, 8, …, a14
3. 18, 15, 12, 9, 6, …, a15
4. 1, -4, -9, -14, -19, …, a9
122
53
60
-39
5.7

67. ACTIVITY 2: Choose the
right one!
From the two choices, select the value
that corresponds to the indicated term of
the arithmetic sequence.
1. a1 = -2, d = 4, a8 = ___
2. a1 = 1, d = 9, a12 = ___
3. a1 = 25, d = -3, a15 = ___
4. a1 = 2.5, d = 2, a20 = ___
26
100
17
25
8
28
109
-17
40.5
-12

68. LET’S
SUMMARIZE
1.What is an
arithmetic
sequence?
2.How do we know
if a sequence is
arithmetic?
3.How do we
determine the

69. THANK
YOU!