MATH 10

1. At the end of this module, you will learn to:
1. Recall concepts on arithmetic sequence.
2. Define and illustrate a geometric sequence.
3. Differentiate a geometric sequence from an
arithmetic sequence.

2. 4, 12, 36, 108, …

3. GEOMETRIC
SEQUENCE
any sequence in which each term after the first
is obtained by multiplying the preceding term
by a fixed number called the common ratio (𝒓)
or a geometric progression

4. COMMON RATIO (𝑟) OF A GEOMETRIC SEQUENCE
divide any term in the
sequence by the term
that precedes it

5. 4, 12, 36, 108, … is a geometric sequence with
a common ratio of 3.
4, 12, 36, 108, …
𝑎1 𝑎2 𝑎3 𝑎4
𝑟 =
12
4
= 3
𝑟 =
36
12
= 3
𝑟 =
108
36
= 3

6. Generate the pattern of the following and determine what
kind of sequence are they .
1. 5, 15, 45, 135, …
2. -8, -17, -26, -35, …
3. 2, 3, 5, 8, …
𝑑 = 15 − 5 = 10
𝑑 = 45 − 15 = 30
𝑑 = 135 − 45 = 90
𝑟 =
15
5
= 3
𝑟 =
45
15
= 3
𝑟 =
135
45
= 3
GEOMETRIC
SEQUENCE
𝑑 = −17 − −8 = −9
𝑑 = −26 − −17 = −9
𝑑 = −35 − −26 = −9
𝑟 =
−17
−8
=
17
18
𝑟 =
−26
−17
=
26
17
𝑟 =
−35
−26
=
35
26
ARITHMETIC
SEQUENCE
𝑑 = 3 − 2 = 1
𝑑 = 5 − 3 = 2
𝑑 = 8 − 5 = 3
𝑟 =
3
2
𝑟 =
5
3
𝑟 =
8
5
NEITHER

7. Nth TERM OF A GEOMETRIC SEQUENCE
𝑎𝑛 = 𝑎1𝑟𝑛−1
where:
𝑎𝑛 = 𝑛𝑡ℎ
term
𝑎1 = 1𝑠𝑡
term
𝑛 =number of terms
𝑟 =common ratio

8. Solve the following problem.
1. Find the common ratio for the geometric sequence 8, 4,
2, 1, ½, …
2. Find the next term in the geometric sequence 3, -9, 27, -
81, . . . .
3. Given 𝑎1 = −4 and 𝑟 = −4, write the first four terms of
the geometric sequence.
4. What is the 11th term in the geometric sequence 2, 6, 18,
…?
5. Given the geometric sequence -18, -6, -2, . . . ,
determine the 26th term.

9. Solve the following problem.
1. What is the 9th term of the geometric sequence 8, 16,
32,…?
2. Find the 5th term of a geometric sequence where the
second term is 108 and the common ratio is 3.
3. In the geometric sequence 4, 12, 36,… which term is
8748?
4. What is the 10th term of a geometric sequence with 32
as the fourth term and 2 as the common ratio?
5. Given the geometric sequence 2, 6, 18,… which term is
118098?

10. GEOMETRIC
MEAN
terms between any two
geometric sequence