MATH 10

1. At the end of this module, you will learn to:
1. Generate patterns.
2. lllustrate an arithmetic sequence.
3. Determine arithmetic means, nth term of an
arithmetic sequence and sum of the terms of a
given arithmetic sequence.

2. Identify the pattern of each sequence then find the next three
terms of each sequence.
1. 2, 5, 8, 11, 14, ___, ___, ___
2. A, B, C, E, F, G, ___, ___, ___
3. 74, 69, 64, 59, ___, ___, ___
4. 1, 4, 16, 64, ____, _____, _____
5. 2, 3, 5, 8, 12, 17, ___, ___, ___
17 20 23
I J K
54 49 44
256 1024 4096
23 30 38
*** To identify the next
terms, patterns should
be determined first.
SEQUENCE
- a list of objects or
numbers arranged in a
definite order

3. Example:
7, 11, 15, 19, … is a sequence. The pattern is
adding 4 to each term.
4th term (𝑎4)
3rd term (𝑎3)
2nd term (𝑎2)
1st term (𝑎1)

4. ARITHMETIC
SEQUENCE
a sequence whose consecutive terms
have a common difference 𝒅 (which can
be identified by subtracting the term by
its previous term)

5. 7, 11, 15, 19, … is an arithmetic sequence with
a common difference of 4.
7, 11, 15, 19, …
𝑎1 𝑎2 𝑎3 𝑎4
11 − 7 = 4
15 − 11 = 4
19 − 15 = 4

6. Determine whether the sequence is an arithmetic
sequence or not by finding its common difference.
1. 4, 10, 16, 22, …
2.17, 19, 21, 24, …
3. 5, 2, -1, -4, …
The common difference 𝑑 = 6, therefore
it is an arithmetic sequence.
There is no common difference, therefore
it is NOT an arithmetic sequence.
The common difference 𝑑 = −3,
therefore it is an arithmetic sequence.

7. Nth TERM OF AN ARITHMETIC SEQUENCE
The nth term of an arithmetic sequence is also called its general
term. It is determined by using the formula:
𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑
where:
𝑎𝑛 = 𝑛𝑡ℎ
term
𝑎1 = 1𝑠𝑡
term
𝑛 =number of terms
𝑑 =common difference

8. Find the next three terms of each sequence.
1. 7, 11, 15, 19, … 2. -5, -7, -9, -11, …
𝑑 = 𝑎2 − 𝑎1
𝑑 = 11 − 7
𝑑 = 4
7, 11, 15, 19, ___, ___, ___
23 27 31
𝑑 = 𝑎2 − 𝑎1
𝑑 = −7 − (−5)
𝑑 = −7 + 5
-5, -7, -9, -11, ___, ___, ___
-13 -15 -17
𝑑 = −2

9. Solve for the indicated term of each arithmetic
sequence.
3. 8th term of the arithmetic sequence 7,11,15,19,…
𝑑 = 𝑎2 − 𝑎1
𝑑 = 11 − 7
𝑑 = 4
𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑
𝑎8 = 7 + 8 − 1 4
𝑎8 = 7 + (7)4
𝑎8 = 7 + 28
𝑎8 = 35

10. Solve for the indicated term of each arithmetic
sequence.
4. 50th term of the arithmetic sequence -5,-7,-9,-11,…
𝑑 = 𝑎2 − 𝑎1
𝑑 = −7 − (−5)
𝑑 = −7 + 5
𝑎𝑛 = 𝑎1 + 𝑛 − 1 𝑑
𝑎50 = −5 + 50 − 1 − 2
𝑎50 = −5 + 49 − 2
𝑎50 = −5 + (−98)
𝑎50 = −103
𝑑 = −2

11. ARITHMETIC SEQUENCE, MEAN & SERIES
a sequence whose
terms have a
common difference
𝒅
𝒂𝒏 = 𝒂𝟏 + 𝒏 − 𝟏 𝒅
the term that lie
between two
nonconsecutive
terms of an
arithmetic
sequence
the sum of terms
of an arithmetic
sequence
𝑺𝒏 =
𝒏
𝟐
𝒂𝟏 + 𝒂𝒏
𝑺𝒏 =
𝒏
𝟐
[𝟐𝒂𝟏 + 𝒏 − 𝟏 𝒅]
ARITHMETIC SEQUENCE ARITHMETIC
MEAN
ARITHMETIC
SERIES