The document provides information about arithmetic sequences and arithmetic series. It defines an arithmetic sequence as a sequence of numbers where each term after the first is obtained by adding the same constant to the previous term. It gives examples of arithmetic sequences and explains how to find the common difference, the nth term of a sequence using the general formula, and how to solve problems involving arithmetic sequences and series. The last paragraph tells a story about how Carl Friedrich Gauss was able to quickly calculate the sum of all numbers from 1 to 100 by recognizing it as an arithmetic series.

1. Arithmetic Sequence
and Arithmetic Series
1

2. 2

3. At the end of the lesson, the learners are expected to:
1) Generate and describe patterns using symbols and mathematical
expressions;
2) Find the next few terms and the nth term of the given sequence;
3) Define and describe an arithmetic sequence;
4) Enumerate the next few terms and the nth term of an arithmetic
sequence;
5) Insert means between two given terms of an arithmetic sequence;
6) Find the sum of the first n terms of an arithmetic sequence; and
7) Solve problems involving arithmetic sequence.
3

4. 4

5. 5

6. 6

7. 1. O T T F F S S E N _?
T E F S
2. R O Y B G I _?
T U V W
3. J F M A M J J A S O N _?
A B C D
4. T Q P H H O N _?
D U H I
Direction: Click the letter of your answer. 7

8. 11

9. 12

10. 1) 80, 40, 20, 10, 5, __?
2) 1, 8, 27, 64, 125, 216, __?
3) 1, 4, 9, 16, 25, 36, 49, __?
343 512 729 810
56 64 72 81
0
1
2
1
5
2
13

11. 4) 1, 2, 3, 5, 8, 13, 21, __?
5) -2, -5, -8, -11, -14, __?
6) 7, 11, 15, 19, 23, 27, __?
-17 -19 -21 -23
29 30 31 32
31 32 33 34
14

12. 19

13. 20

14. 21

15. 22

16. 23

17. 24

18. 25

19. 26

20. 27

21. 28

22. 29

23. 30

24. 31

25. An ordered list of numbers, such as
10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, …
is called a sequence.
A number sequence is a list of numbers having a
first number, a second number, a third number, and
so on, called the terms of the sequence.
32

26. FINITE AND INFINITE
SEQUENCES
• 10, 15, 20, 25, 30
• There are only a
finite number of
terms.
• They have a last
term.
• 11, 22, 33, 44, 55,
…
• There are infinite
number of terms.
• They do not have a
last term.
33

27. Sequences are usually given by stating their
general or nth term.
NUMBER SEQUENCE
Example:
Consider the sequence given 𝒂 𝒏 = 𝟑𝒏 + 𝟐
The first five terms of the sequence are
𝒂 𝟏 = 𝟑 𝟏 + 𝟐 = 𝟑 + 𝟐 = 𝟓,
𝒂 𝟐 = 𝟑 𝟐 + 𝟐 = 𝟔 + 𝟐 = 𝟖,
𝒂 𝟑 = 𝟑 𝟑 + 𝟐 = 𝟗 + 𝟐 = 𝟏𝟏,
𝒂 𝟒 = 𝟑 𝟒 + 𝟐 = 𝟏𝟐 + 𝟐 = 𝟏𝟒, 𝒂𝒏𝒅
𝒂 𝟓 = 𝟑 𝟓 + 𝟐 = 𝟏𝟓 + 𝟐 = 𝟏𝟕
34

28. a. 𝒂 𝒏 = 𝟐𝒏 𝟐
for 𝟏 ≤ 𝒏 ≤ 𝟓
b. 𝒂 𝒏 = 𝟐𝒏 − 𝟏 for 𝟏 ≤ 𝒏 ≤ 𝟓
c. 𝒂 𝒏 = (−𝟏) 𝒏
(𝒏 − 𝟑) 𝟐
for 𝟏 ≤ 𝒏 ≤ 𝟒
List all the indicated terms of each finite
sequence.
NUMBER SEQUENCE
35

29. 37

30. QUESTION:
What do you observe in the following pictures
38

31. 39

32. Joey’s school offered him a scholarship grant of ₱
3,000.00 when he was in Grade 7 and increased
the amount by ₱ 500 each year till Grade 10. The
amounts of money (in ₱) Joey received in Grade 7,
8, 9, and 10 were respectively:
3000, 3500, 4000, and 4500
Each of the numbers in the list is called a term.
Note: We find that the succeeding terms are
obtained by adding a fixed number.
40

33. In a savings scheme, the amount triples after every
7 years. The maturity amount (in ₱) of an
investment of ₱ 6,000.00 after 7, 14, 21 and 28
years will be, respectively:
18000, 54000, 162000, 486000
Note: We find that the succeeding terms are
obtained by multiplying with a fixed number.
41

34. NUMBER PATTERNS
The number of unit squares in a square with sides
1, 2, 3, 4, ... units are respectively 1, 4, 9, 16, ....
Note: We can observe that 1=12, 4=22, 9=32, 16=42, ...
Thus, the succeeding terms are squares of consecutive
numbers.
42

35. 43

36. ARITHMETIC SEQUENCE
Consider the following lists of numbers :
2, 4, 6, 8, 10, ....
15, 12, 9, 6, 3, ....
-5, -4,-3, -2, -1....
6, 6, 6, 6, 6, 6, ....
each term is obtained by adding 2 to
the previous term
each term is obtained by adding -3
to the previous term
each term is obtained by adding 1 to
the previous term
each term is obtained by adding 0 to
the previous term
44

37. 45

38. 1) 23, 38, 53, __ , 83, 98
2) 45, 37, __ , 21, 13, 5
3) -13, -6, __ , 8, 15, 22
27 28 29 30
0 1 -2 -3
63 68 73 78
46

39. 4) __ , 23, 32, 41, 50, 59
5) -12, -7, -2, 3, 8, ___
6) 10, __ , 32, 43, 54, 65
10 11 12 13
21 23 25 27
10 12 14 16
47

40. The constant is called the common difference, denoted with
the letter d, referring to the fact that the difference between
two successive terms yields the constant value that was
added. To find the common difference, subtract the first term
from the second term.
In mathematics, an arithmetic sequence or arithmetic
progression is a sequence of numbers where each term
after the first term is obtained by adding the same
constant (always the same).
52

41. Examples:
5, 10, 15, 20, 25, …
3, 7, 11, 15, …
20, 18, 16, 14, …
-7, -17, -27, -37, …
d=5
d=4
d=-2
d=-10
53

42. 2, 5, 8, 11, 14 …
In this sequence, notice that the common difference is
3.
In order to get to the next term in the sequence, we
will add 3; so, a recursive formula for this sequence
is:
31  nn aa
The first term in the sequence is a1 (sometimes just
written as a).
Example:
54

43. 2, 5, 8, 11, 14 …
+3 +3 +3 +3
Every time you want another term in the said sequence of
numbers, you have to add d. This means that the second
term was the first term plus d. The third term is the first
term plus d plus d (added twice). The fourth term is the
first term plus d plus d plus d (added three times). So,
you can see to get the nth term we have to take the first
term and add (n - 1) times d.
𝑑 = 3
 dnaan 1
𝑎 = 2
  1715231626 a 55

44. To find any term of an arithmetic sequence:
where:
a1 is the first term of the sequence,
d is the common difference,
n is the number of the term to find.
ARITHMETIC SEQUENCE
56

45. Find the nth term of the arithmetic sequence
5, 8, 11, 14, …
We know that 𝒂 𝟏 = 𝟓 𝒅 = 𝟑
Substituting in the formula, we obtain
𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑
𝑎 𝑛 = 5 + 𝑛 − 1 3
𝑎 𝑛 = 5 + 3𝑛 − 3
𝒂 𝒏 = 𝟐 + 𝟑𝒏
57

46. Find the 18th term of the arithmetic sequence
21, 24, 27, 30, 33, …
Note that 𝒂 𝟏 = 𝟓, 𝒅 = 𝟑, 𝒏 = 𝟏𝟖. Using the formula
for the general term of an arithmetic sequence, we have
𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑
𝑎 𝑛 = 21 + 18 − 1 3
𝑎 𝑛 = 21 + (17)3
𝒂 𝒏 = 𝟕𝟐
Thus, 72 is the 18th term of the sequence.
58

47. In the arithmetic sequence 10, 16, 22, 28, 34, … , which
term is 124?
The problem asks for n when 𝑎 𝑛 = 124. From the given
sequence 𝒂 𝟏 = 𝟏𝟎, 𝒅 = 𝟔, 𝒂 𝒏 = 𝟏𝟐𝟒. Substitute these
values in the formula to get
𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑
124 = 10 + 𝑛 − 1 6
124 = 10 + 6𝑛 − 6
120 = 6𝑛
𝟐𝟎 = 𝒏 Thus, 124 is the 20th term.
59

48. Find the first term of the arithmetic sequence with a
common difference of 6 and whose 20th term is 965?
Since 𝒂 𝟐𝟎 = 𝟗𝟔𝟓, 𝒅 = 𝟔, 𝒏 = 20 , we have
𝑎20 = 𝑎1 + 𝑛 − 1 𝑑
965 = 𝑎1 + 20 − 1 6
965 = 𝑎1 + 19 6
965 = 𝑎1 + 114
𝑎1 = 965 − 114 = 851
Thus, 851 is the first term.
60

49. Find the 15th term of the arithmetic sequence whose first
term is 7 and whose 5th term is 19.
Find the common difference, d by substituting 𝑎1 =
7 and, 𝑎5 = 19 . Thus,
𝑎5 = 𝑎1 + 𝑛 − 1 𝑑
19 = 7 + 5 − 1 𝑑
19 = 7 + 4𝑑
12 = 4𝑑
3 = 𝑑
Now use 𝒂 𝟏 = 𝟕, 𝒅 = 𝟑, 𝒏 = 𝟏𝟓
𝒂 𝟏𝟓 = 𝟕 + 𝟏𝟓 − 𝟏 𝟑 = 𝟒𝟗
Thus, the 15th term is 49.
61

50. Insert two arithmetic means between 8 and 17.
Our task here is to find 𝑎2 and 𝑎3 such that 8, 𝑎2, 𝑎3,
17 form an arithmetic sequence.
𝑎1 = 8 𝑎4 = 17
𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑
17 = 8 + 4 − 1 𝑑
17 = 8 + 3𝑑
15 = 3𝑑
5 = 𝑑
We need 𝑎2 and 𝑎3 . Hence, 𝑎2 = 8 + 3 = 𝟏𝟏 and
𝑎3 = 8 + 3 + 3 = 𝟏𝟒
62

51. A conference hall has 33 rows of seats. The last row
contains 80 seats. If each row has two fewer seats than
the row behind it. How many seats are there in the first
row?
We know that 𝒂 𝟑𝟑 = 𝟖𝟎, 𝒅 = 𝟐, 𝒏 = 33 . Using the formula
for the general term of an arithmetic sequence, we have
𝑎33 = 𝑎1 + 𝑛 − 1 𝑑
80 = 𝑎1 + 33 − 1 2
80 = 𝑎1 + 32 2
80 = 𝑎1 + 64
𝑎1 = 80 − 64 = 16
Therefore, there are 16 seats in the first row.
63

52. How many numbers between 7 and 500 are divisible by
5?
The common difference, d, is 5. Since we want numbers
that are divisible by 5, then we let 𝒂 𝟏 = 𝟏𝟎, 𝒂 𝒏 = 𝟓𝟎𝟎.
Substitute these values into the general term.
𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑
500 = 10 + 𝑛 − 1 5
500 = 10 + 5𝑛 − 5
495 = 5𝑛
𝟗𝟗 = 𝒏 There are 99 numbers between 7 and 500 that are
divisible by 5.
64

53. 65

54. 1. 1, 2, 3, 4, … and 1+2+3+4+…
2. 2, 4, 6, 8, … and 2+4+6+8+…
3. 5, 10, 15, … and 5+10+15+…
4. 3, 7, 11, 15, … and 3+7+11+15+…
5. 4, 8, 12, 16, … and 4+8+12+16+…
Each indicated sum of the terms of an
arithmetic sequence is an arithmetic series.
66

55. Do you know this?
The story is told of a grade school teacher In the 1700’s
that wanted to keep her class busy while she graded
papers so she asked them to add up all of the numbers
from 1 to 100. These numbers are an arithmetic sequence
with common difference 1. Carl Friedrich Gauss was in
the class and had the answer in a minute or two
(remember no calculators in those days). This is what he
did:
1 + 2 + 3 + 4 + 5 + . . . + 96 + 97 + 98 + 99 + 100
sum is 101
sum is 101
67

56. The formula for the sum of n terms is:
 nn aa
n
S  1
2
n is the number of terms so
𝒏
𝟐
would be the number of pairs
Let’s find the sum of 1 + 3 +5 + . . . + 59
ARITHMETIC SERIES
first term last term
68

57.  nn aa
n
S  1
2
Let’s find the sum of 1 + 3 +5 + . . . + 59
 12 nThe common difference is 2 and
the first term is 1, so:
Set this equal to 59 to find n. Remember n is the term number.
𝟐𝒏 − 𝟏 = 𝟓𝟗 𝒏 = 𝟑𝟎 So there are 30 terms to sum up.
  900591
2
30
30 S
first term last term
69

58. To find the sum of a certain number of
terms of an arithmetic sequence:
where:
Sn is the sum of n terms (nth partial sum),
a1 is the first term,
an is the nth term.
70

59. To find the sum of a certain number of
terms of an arithmetic sequence:
where:
Sn is the sum of n terms (nth partial sum),
a is the first term,
n is the “rank” of the nth term
d is the common difference
71

60. Find the sum of the first ten positive integers.
a1 = 1 n = 10
Illustrative Example
a10 = 10
𝑆10 =
𝑛
2
(𝑎1 + 𝑎 𝑛)
𝑆10 =
10
2
(1 + 10)
𝑆10 = 5 (11)
𝑆10 = 𝟓𝟓
72

61. Find the sum of the first 15 terms of the arithmetic
sequence if the first term is 11 and the 15th term
is 109.
a1 = 11 n = 15
Illustrative Example
a15 = 109
𝑆15 =
𝑛
2
(𝑎1 + 𝑎 𝑛)
𝑆15 =
15
2
(11 + 109)
𝑆15 =
15
2
(120)
𝑆15 = 𝟗𝟎𝟎
73

62. Find the sum of all the odd integers from 1 to
99.
a1 = 1 d = 2
Illustrative Example
Here, a10 = 99
𝑆50 =
𝑛
2
(𝑎1 + 𝑎 𝑛)
𝑆50 =
50
2
(1 + 99)
𝑆50 = 25 (100)
𝑆50 = 𝟐, 𝟓𝟎𝟎
𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑
99 = 1 + 𝑛 − 1 2
99 = 1 + 2𝑛 − 2
100 = 2𝑛
50 = 𝑛
74

63. In a classroom of 40 students, each student counts off
by fours (i.e. 4, 8, 12, 16, …). What is the sum of the
students’ numbers?
a1 = 4 d =4
Illustrative Example
Here, n=40
𝑆40 =
𝑛
2
(𝑎1 + 𝑎 𝑛)
𝑆40 =
40
2
(4 + 160)
𝑆40 = 20 (164)
𝑆40 = 𝟑, 𝟐𝟖𝟎
𝑎 𝑛 = 𝑎1 + 𝑛 − 1 𝑑
𝑎 𝑛 = 4 + 40 − 1 4
𝑎 𝑛 = 4 + 156
𝑎 𝑛 = 160
75

64. 76

65. 77

66. Practice Exercises
Determine if the given sequence is an arithmetic
sequence or not.
1. 6, 8, 10, 12, …
2. -11, -10, -9, -8, …
3. 8, 11, 14, 17, …
4. 5, 15, 45, 135, …
5. 6, 11, 17, 21, …
Arithmetic Sequence NOT
Arithmetic Sequence
Arithmetic Sequence
Arithmetic Sequence
Arithmetic Sequence
NOT
NOT
NOT
NOT
Click the figure which corresponds to your answer. 78

67. Practice Exercises
Solve the following problems.
6. In the arithmetic sequence , 2, 5, 8, 11, …, what is the 30th
term?
7. In the arithmetic sequence 8, 5, 2, -1, …, what is the 15th term?
8. In the arithmetic sequence with 12 as the first term and the
common difference is -3, what is the 17th term?
87
-28
88 89 90
-30 -32 -34
-36 -37 -38 -39
Click the figure which corresponds to your answer. 79

68. Practice Exercises
Solve the following problems.
9. In the arithmetic sequence 23, 30, 37, 44, …, what is the 14th
term?
10. In the arithmetic sequence 6, 12, 18, …, what is the 29th term?
110
170
112 114 116
172 174 176
Click the figure which corresponds to your answer. 80

69. Practice Exercises
Find the arithmetic mean of the following numbers.
11) 4 and 16
12) 19 and 35
13) 13 and 25
14) -22 and 8
15)102 and 1002
-6
8 10 12
26 27 28 29
6
15 17 2119
550
-7
552
-8
554
-9
556
Click the figure which corresponds to your answer. 81

70. Practice Exercises
Solve the following problems.
16. What is the sum of the first 100 positive odd integers?
17. What is the sum of the first 50 positive even integers?
18. What is the sum of the first 30 positive multiples of 8?
3730
11000 12000 13000
2500
3710
2550 2600 2650
3720
10000
3740
Click the figure which corresponds to your answer. 82

71. Practice Exercises
19. Aris takes a job, starting with an hourly wage of ₱
350.00 and is promised a raise of ₱ 5.00 per hour every
two months for 5 years. At the end of 5 years, what
would be Aris’ hourly wage?
20. Find the sum of all two-digit even natural numbers
₱ 485 ₱ 490 ₱ 495 ₱ 500
2410 2420 2430 2440
Click the figure which corresponds to your answer. 83

72. 96

73. Assessment
Determine if the sequence is an arithmetic
sequence or not.
1. 9, 11, 13, 15, …
2. 6, 11, 16, 21, …
3. 1, 2, 4, 8, 16, …
4. -4, 2, 8, 14, …
5. 1, 8, 27, 64, …
97

74. Assessment Solve the following problems.
6. What is the 11th term in the sequence 6, 9, 12, 15,… ?
7. What is the 24th term of the sequence 16, 19, 22, … ?
8. What is the 25th term of the sequence 12, 9, 6, … ?
9. What is the 30th term in the arithmetic sequence with a
first term of 15 and a common difference of 5?
10. What is the 10th term of the arithmetic sequence with
a first term of 75 and a common difference of -8?
98

75. Assessment
11. Insert the arithmetic mean of 8 and 28.
12. Insert two arithmetic means between 16 and 31.
13. Insert two arithmetic means between 21 and 33.
14. Insert three arithmetic means between 11 and 35.
15. Insert three arithmetic means between 48 and 84.
99

76. Assessment Solve the following problems.
16. Joan started a new job with an annual salary of ₱ 150
000 in 2007. If she receives a ₱ 12 000 raise each year, how
much will her annual salary be in 2017?
17. A stack of telephone poles has 30 poles in the bottom
row. There are 29 poles in the second row, 28 in the next
row, and so on. How many poles are there in the 26th row?
18. Josh spent ₱ 150 on August 1, ₱ 170 on August 2. ₱ 190
on August 3, and so on. How much did Josh spend on
August 31?
100

77. Assessment
19. An object is dropped from a jet plane and falls 32
feet during the first second. If during each successive
second, it falls 40 feet more than the distance in the
preceding second, how far does it fall during the
eleventh second?
20. What is the seating capacity of a movie house with
40 rows of seats if there are 25 seats in the first row,
28 seats in the second row, 31 in the third row, and so
on?
101

78. 107

79. • Acelajado, Maxima J. (2008). New High School Mathematics II Second
Edition. Makati City: Diwa Learning Systems, Inc.
• Callanta, Melvin M., et al. (2015). Mathematics – Grade 10 Learner’s
Module. Pasig City: Department of Education.
• Orines, Fernando B., et al. (2008). Next Century Mathematics
(Intermediate Algebra) Second Edition. Quezon City: Phoenix Publishing
House, Inc.
• Oronce, Orlando A., Mendoza, Marilyn O. (2010). E-Math II. Manila: Rex
Book Store, Inc.
• http://www.google.com.ph
(Some of the pictures used in this presentation were taken from the said site)
• http://www.slideshare.com
(Some of the examples and exercises of arithmetic sequence and arithmetic
series used in this presentation were taken from the said site)
• https://www.youtube.com/watch?v=HlZky0FL6ck
(The video used in this presentation was taken from the said site)
108

80. 109