Module 3 covers the concepts of arithmetic sequences, including determining the nth term, finding arithmetic means, and calculating the sum of terms. It includes lessons and examples to practice identifying arithmetic sequences and applying formulas to solve for specific terms. The module concludes with exercises and a problem-solving section related to the concepts taught.

3. Welcome to Module 3!
This module is divided in to three lessons:
o Lesson 1 – nth term of an Arithmetic Sequence
o Lesson 2 – Arithmetic Means
o Lesson 3 – Sum of the Terms of a Given Arithmetic
sequence of a given Arithmetic
Sequence
Reminder: Use this module with care.
Next

4. Next
After going through this
module, you are expected
to:
1. determine the nth
term of an arithmetic
sequence;
2. define arithmetic mean;
3. find the sum of the terms
of a given arithmetic
sequence.

5. Next

6. Next

7. In Lessons 1 & 2, you have learned to illustrate sequences and
give the nth term or the rule for a particular sequence. These are
set of objects or numbers arranged in sequence order.
Example:
Write the first 5 terms of a sequence given the nth term
an = 3n – 1.
Let a1, a2, a3, a4, and a5 be the first 5 terms. Thus,
n = 1, 2, 3, 4, and 5.
a1 = 3(1) - 1 = 2
a2 = 3(2) – 1 = 5
a3 = 3(3) – 1 = 8
a4 = 3(4) – 1 = 11
a5 = 3(5) – 1 = 14.
The first 5 terms of the nth term an = 3n – 1 are 2, 5, 8, 11,and 14.
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8. This lesson focuses on the nth term of an arithmetic sequence, arithmetic
means,
and the sum of the terms of a given arithmetic sequence. Read the
problem
below and answer the questions that follow.
Try this:
Determine if the following series of numbers are arithmetic sequences or
not. If the given is an arithmetic sequence, write A on the space provided.
If not, write N.
______ 1. 2, 4, 6, 8, 10, 12, 14
______ 2. 5, 4, 7, 9, 11, 10, 8
______ 3. 25, 28, 31, 34, 37, 40, 43
______ 4. 14, 15, 17, 17, 19, 20, 21
______ 5. 124, 126, 128, 130, 132, 134, 136
Compare your answers with those found in the Answer Key.
Next

9. Determine if the following series of numbers are arithmetic
sequences or not. If the given is an arithmetic sequence,
write A on the space provided. If not, write N.
A 1. 2, 4, 6, 8, 10, 12, 14
N 2. 5, 4, 7, 9, 11, 10, 8
A 3. 25, 28, 31, 34, 37, 40, 43
N 4. 14, 15, 17, 17, 19, 20, 21
A 5. 124, 126, 128, 130, 132, 134, 136
If you got 3 to 5 correct answers, you’re doing great. You
can proceed to the next lesson. If you got only 2 or less, read
Module 2 again and after doing that, try to solve the
exercises given above again.
Next

10. EXAMPLE 1
Consider the sequence 2, 6, 10, 14, . . . . What is the 12th term in the given sequence?
Follow the steps below to solve this problem.
STEP 1 Find the common difference.
6 – 2 = ____
10 – 6 = ____
14 – 10 = ____
The common difference is: d = 4.
STEP 2 Determine the 1st term in the given arithmetic sequence.
The 1st term in the given arithmetic sequence is 2. This means that a1= 2.
STEP 3 Find the symbol for the unknown term in the sequence.
You are asked for the 12th term in the given arithmetic sequence. Thus, the symbol for the
unknown term is a12.
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11. STEP 4 Write the equation or the number sentence for the unknown term in the sequence.
The equation for a12 is:
a12 = a1+ (12 – 1) d
a12 = a1+ 11d
STEP 5 Substitute the values in the equation and solve for the answer.
From Step 1, d = 4 and from Step 2, a1 = 2. Thus we have
a12 = a1+ 11d
a12 = 2 + 11 (4) = 46
This means that the 12th term of the arithmetic sequence 2, 6, 10, 14, . . . . . .is 46.
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12. REMEMBER:
To solve for the nth term in an arithmetic sequence, we use the formula:
an = a1 + (n–1)d where an = nth term
a1 = 1st term
n = number of terms
d = common difference
♦ We follow the following steps in using the equation above:
STEP 1 Find the common difference.
STEP 2 Determine the 1st term in the given arithmetic sequence.
STEP 3 Find the symbol for the unknown term in the sequence.
STEP 4 Write the equation for the unknown term in the sequence.
STEP 5 Substitute the values in the equation and solve for the result.
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13. REMEMBER:
One of the common tasks in studying arithmetic sequence is finding the arithmetic means. The
arithmetic means are the terms between any two non-consecutive terms of an arithmetic
sequence.
Examples: a.) Insert 2 arithmetic means between 4 and 19.
Solution: Since we are required to insert 2 terms, then there will be 4 terms in all.
Step 1: Let a1 = 4 and a4 = 19. Then we will insert a2, a3 as shown below:4, a2, a3, 19
Step 2: Get the common difference. Let us use a4 = a1 + 3d
Step 3: Substitute the given values of a4 and a1. That is 19 = 4 + 3d.
Step 4: Solve for d.
19 – 4 = 3d
15 = 3d
5 = d
Step 5: Using the value of d, we can now get the values of a2, and a3. Thus,
a2 = 4 + 5(1) = 9
a3 = 4 + 5(2) = 14
Answer: The 2 arithmetic means between 4 and 19 are 9 and 14.
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14. Shown in the table is a list of the number of bottles of juice drinks
Jaine-Anne sold from Monday to Saturday.
PROBLEM: How many bottles of soft drinks will Jaine-Anne
sell in 10 days?
From the series of numbers, find out the following:
1. Is the series an arithmetic sequence?
2. Do the numbers in the series have a common difference? If yes, what is the common difference or d?
3. What is the first term or a1?
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15. From the series of numbers, find out the following:
1. Is the series an arithmetic sequence?
2. Do the numbers in the series have a common difference? If yes,
what is the common difference or d?
3. What is the first term or a1?
Next
Day No. of Bottles
1 20
2 70
3 120
4 170
5 220
6 270

16. The symbol for sum is S. then for the sum of:
✓ two terms is S2
✓ three terms is S3
✓ four terms is S4
✓ five terms is S5
✓ six terms is S6 ,.. What is the symbol for the sum of 7 terms? 10 terms? 12 terms?
If your answer are S7, S10 and S12 , you are correct!
Here’s the formula to find the sum of the terms of an arithmetic sequence:
Next
1

17. Given:
a1 = 20
n = 10
Solve for S6.
Formula:
Solution:
S10 = [2(20) + (10-1)50]
=5 [40 + (9)50]
=5 [40 + 450)
=5 (490)
=2,450 Next
10
2
Answer: There are 2,450 bottles of soft drinks will
Jaine-Anne sell in 10 days.
1

18. Find
the nth
term
You
Mean
to Me
LEVEL 1 LEVEL 2
Sum
it up
LEVEL 3

19. A. 35
B. 36
C. 37
1. What is the next term in
the arithmetic sequence
3, 11, 19, 27, ____?

22. A. 22
B. 24
C. 26
2. What is the 6th term in
the arithmetic sequence
sequence 4, 8, 12,…?

25. A. 30
B. 35
C. 40
3. What is the 8th term in
arithmetic sequence
5, 10, 15,…?

28. A. -53
B. 53
C. 35
4. What is the 10th term of
the arithmetic sequence
10, 3, -4, - 11, …?

31. 5. What is the 15th term of
the arithmetic sequence
3, 6, 9, 12,…?
A. 35
B. 45
C. 50

32. Next

35. A C
B
30 40 50
1. What is the arithmetic mean between 20
and 60?

38. A C
B
34 35 36
2. The arithmetic mean between two terms in an
arithmetic sequence is 24. If one of these terms is 13,
find the other term.

41. A C
B
31 32 33
3. If three arithmetic means are inserted between 21 and
45, find the middle arithmetic mean.

44. 4. If five arithmetic means are inserted between 11 and 47,
what is the 2nd arithmetic mean?
A C
B
23 29 35

47. A C
B
23 and 7 53 and -7 43 and 7
5. What are the first and last terms of an arithmetic
sequence when the arithmetic means are 33 and 13?

51. A
C
B
182
172
162

54. A
C
B
219
319
419

57. A
C
B
120
130
140

60. A
C
B
185
175
165

63. A
C
B
219
220
221

66. 1. What are the steps in finding the nth
term of an arithmetic sequence?
Next
2. What is an arithmetic mean?
3. What is the formula to find the
sum of the terms in an
arithmetic sequence?

67. Next

68. Arithmetic or Not?
1)3, 7, 11, 15, 19 Yes No

69. Arithmetic or Not?
2)4, 16, 64, 256 Yes No

70. Arithmetic or Not?
3) 48, 24, 12, 6, 3, ...
Yes No

71. Arithmetic or Not?
4) 1, 4, 9, 16, 25, 36
Yes No

72. Arithmetic or Not?
5) 9.5, 7.5, 5.5, 3.5,…
Yes No

73. 1) Find the 10th term in the arithmetic
sequence with –9 as the 1st term and 4
as the common difference
27
23
25

74. 2) What is the 12th term in a sequence if the
first term is 5 and the common difference is
5?
50
70
60

75. 3) Find the 6th term in the sequence 3,
6, 9, . . .
16
18
17

76. 4) 5th term in the sequence if the 1st term
is 64 and the common difference is 4
80
86
83

77. 1) What is the arithmetic mean between 50 and 100?
95 85 75

78. 2)If three arithmetic means are inserted between 21 and 45,
find the middle arithmetic mean.
23 33 43

79. 3)The arithmetic mean between two terms in an arithmetic
sequence is 24. If one of these terms is 13,
find the other term.
35 37 39

80. 4)If five arithmetic means are inserted between 10 and 46,
what is the 3rd arithmetic mean?
28 29 30

81. 1)integers from 1 to 50
1275 1277
1276

82. 2) odd integers from 1 to 100
2525 2575
2500

83. 3) even integers between
1 and 101
2050 2550
2150

84. 4) first 25 terms of the arithmetic
sequence 4, 9, 14, 19, 24, ...
1600 1800
1700

85. 5.) first 20 terms of the arithmetic
sequence –16, –20, –24, …
-1090 -1070
-1080

86. Your score is 
12
Next

87. Congratulations!

89. hilda.dragon002@deped.gov.ph
Send your questions to Teacher Hilda ☺