This document provides a strategic intervention material on illustrating geometric sequences for mathematics learners in 10th grade. It contains various activities to distinguish between arithmetic and geometric sequences, identify the common ratio of geometric sequences, find the nth term and sum of finite and infinite geometric sequences, and determine geometric means. The activities include examples of dividing terms to find ratios, identifying common ratios, stating whether sequences are geometric or not, finding missing terms, using the nth term formula, and assessing understanding. The document was prepared by a teacher and submitted for approval to help students master the key skills of working with geometric sequences.

1. STRATEGIC INTERVENTION MATERIAL
MATHEMATICS 10
PREPARED BY:
ALMA EDQUID BAJA
T-2
Department of Education
Region III
Schools Division of Zambales
GUISGUIS NATIONAL HIGH SCHOOL
Guisguis, Sta. Cruz, Zambales

2. LEAST MASTERED SKILLS
Illustrates Geometric
Sequence
Sub-Tasks:
• Distinguish between arithmetic and
geometric sequences.
• Find the common ratio of a geometric
sequence.
• Determine the geometric means and nth
term of a geometric sequence.
• Find the sum of the terms of a given finite
and infinite geometric sequence.

3. A Geometric Sequence
is sequence where each
term after the first is
obtained by multiplying
the preceding term by a
nonzero constant called
the common ratio.
How are
sequences used
to model and
solve many
mathematical
ideas and real-
life situations?
You need the concept
of ratio in order to
understand the
geometric sequences.
We will explore that
sequence by doing
some activities. Are
you ready? Lets go!

4. Activity #01
Divide and Conquer
Find the ratio of the second
number to the first number.
1. 2, 8 6. -49, 7
2. -3, 9 7. ¼, ½
3. 1, ½ 8. 𝒂 𝟐
, 𝒂 𝟑
4. -5, -10 9. k – 1, k
5. 12, 4 10. 3m, 3mr

5. Activity #02
Give Me My Ratio!
Identify the
common ratio of the
following geometric
sequences.
1. 4, 12, 36, . . .
2. 3, 6, 12, 24, . . .
3. 36, 18, 9, . . .
4. -3, 12, -48, . . .
5. -5, -25, -125, . . .
Thus, in the geometric
sequence 2, 4, 8, 16,
32, . . . , the common
ratio is 2 since
𝟑𝟐
𝟏𝟔
= 2

6. The next activity will
test whether you can
identify geometric
sequences or not.
Goodluck!
Activity #03:
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 2, 5 2, 3 2, 2
3. 5, -10, 20, -40, . . .
4. 1, 0.6, 0.36, 0.216, .
. .
5. 4, 0, 0, 0, 0, . . .

7. Activity # 04:
Missing You
Find the missing terms in
each geometric sequence.
1. 3, 12, 48, ___, ___
2. ___, ___, 32, 64, 128, .
.
3. 120, 60, 30, ___, ___
4. 5, ___, 20, 40, ___,
5. ___, 4, 12, 36, ___

8. Remember:
The nth term of a
geometric sequence is
𝒂 𝒏 = 𝒂 𝟏 𝒓 𝒏−𝟏
where
𝒂 𝟏 = the first term
𝒂 𝒏 = the nth term
r = the common ratio
Find the tenth term of the
geometric sequence 2, 4, 8, . . .
Solution: 𝑎1 = 2; r = 2; n = 10
Formula: 𝑎 𝑛= 𝑎1 𝑟 𝑛−1
𝑎10= 2(2)10−1
𝑎10 = 2(2)9
= 1024 – the tenth term
Activity #05: There’s More
on Geometric Sequences
Use the nth term of a geometric
sequence 𝒂 𝒏 = 𝒂 𝟏 𝒓 𝒏−𝟏
to answer
the following questions.
1. What is the 5th term of the
geometric sequence 3, 6,
12, . . .?
2. Find the 6th term of a
geometric sequence where
the 2nd term is 6 and the
common ratio is 2.
3. Find the tenth term of the
geometric sequence 2, -6,
18, . . .

9. Activity #06
Finding Geometric
Means
Inserting a certain number
of terms between two given
terms of a geometric
sequence is an interesting
activity in studying
geometric sequences.
We call the terms between
any two given terms of a
geometric sequences the
Geometric Means.
Example:
Insert 2 geometric
means between 5 and 625
Solution:
Let 𝑎1= 5 and 𝑎4 = 625. we
will insert 𝑎2, 𝑎𝑛𝑑 𝑎3.
Since 𝑎4 = 𝑎1 𝑟3
, then 625 =
5𝑟3
.
Solving for the value of r, we
get 125 = 𝑟3
or r = ±5
We obtained two values of r,
so we have two geometric
sequences.

10. If r = 5, the geometric means
are 𝑎2= 5 5 1
= 25 𝑎3= 5 5 2
=
125.
Thus, the sequence is 5, 25, 125,
625.
If r = -5, the geometric means
are 𝑎2= 5 −5 1
= -25 𝑎3=
5 5 2
= 125.
Thus, the sequence is 5, -25,
125, -625.
Find the indicated number of
geometric means between
each pair of numbers.
1. 16 and 81 (3)
2. 256 and 1 (3)
3. -32 and 4 (2)
4.
1
3
and
64
3
(1)
5. 2xy and 16𝑥𝑦4
(1)
Was knowing the nth term of a
geometric sequence helpful in
finding geometric means?

11. Activity # 07: Sum of
Terms in a Geometric
Sequence
Answer the following:
1. Find the sum of the first 5
terms of 4, 12, 36, . . .
2. Find the sum of the first 6
terms of 3, -6, 12, -24, . . .
3. . Find the sum of the first
6 terms of -3, 3, -3, 3, . . .
4. Find the sum of the first 7
terms of -3, 3, -3, 3, . . .
5. Find the sum of the first 8
terms of -
3
4
,
3
4
,
3
4
,
3
4
, . . .
Formula:
Sum of Finite Geometric
Sequence
𝑆 𝑛 =
𝑎1(1−𝑟 𝑛)
1−𝑟
or
𝑎1−𝑎 𝑛 𝑟
1−𝑟
where: 𝑆 𝑛 = the sum
𝑎1 = the first term
r = the common ratio, r≠1.
𝑆 𝑛 = n 𝑎1 if r = 1
Sum of Infinite Geometric
Sequence
S =
𝑎1
1−𝑟
where: 𝑆 𝑛 = the sum
𝑎1 = the first term
r = the common ratio, /r/ <
1.

12. Assessment #01:
How well do you know
me?
Determine whether each
sequence is arithmetic,
geometric, or neither.
If the sequence is
arithmetic, give the
common difference; if
geometric, give the
common ratio.
1. 6, 18, 54, 162, . .
.
2. 4, 10, 16, 22, . . .
3. 1, 1, 2, 3, 5, 8, .
. .
4. 625, 125, 25, 5, . .
.
5. 5, 8, 13, 21, 34, .
. .

13. Assessment #02:
Do You Remember Me?
Answer the following.
1. Find the fifth term of
the geometric sequence
5, 10, 20, . . .
2. Find the geometric
mean between 3 and 12.
3. Insert two geometric
mean between 3 and 81.
4. Find the sum of the
first 8 terms of the
geometric sequence -5,
5, -5. 5, . . .
5. Find the sum of the
first 7 terms of the
geometric sequence -5,

14. State whether the given
sequence is arithmetic
or geometric. Then give
the next term of the
sequence.
1. 8, 16, 24, 32, . . .
2.
1
3
,
1
9
,
1
27
,
1
81
, . . .
3. 5, 10, 15, 20, . . .
4. 5, 10, 20, 80, . . .
5. 9, 19, 29, 39, . . .Mastery Points!
Can You
• Distinguish between arithmetic and geometric
sequences?
• Identify the common ratio of a geometric
sequence?
• Identify the geometric means and find the nth
term of a geometric sequence?
• Find the sum of the terms of a given finite and
infinite geometric sequence.

15. - Mathematics Learners
Module-Grade10 (DEPED) p.
26 - 47
- Intermediate Algebra -
Soledad Jose-Dilao, Ed.D.
Julieta G. Bernabe, Published
by JTW Corporation.
- Intermediate Algebra,
Corazon Alano, et.al,
Abiva Publishing House,
Inc.
- http://www.goggle.com

16. Activity 1:
Divide and Conquer
1. 4 6. -1/7
2. -3 7. 2
3. ½ 8. a
4. 2 9.
𝒌
𝒌−𝟏
5. 1/3 10. r
Activity 2:
Give Me My
Ratio!
1. r = 3
2. r = 2
3. r = ½
4. r = - 4
5. r = 5
Activity 3:
I’ll TellYouWhatYou Are
1.Geometric Sequence
2. Not
3. Geometric Sequence
4. Geometric Sequence
5. Not
Activity 4:
MissingYou
1. 192, 768
2. 8, 16
3. 15,
𝟏𝟓
𝟐
4. 10, 80
5.
𝟒
𝟑
, 108

17. Activity 5:
There’s More On
Geometric Sequences
1. 𝒂 𝟓 = 𝟒𝟖
2. 𝒂 𝟔 = 96
3. 𝒂 𝟏𝟎 = −𝟑𝟗𝟑𝟔𝟔
Activity 7:
Sum of Terms in a
Geometric
Sequence
1. 484
2. -63
3. 0
4. -3
5. 6
Activity 6:
Finding Geometric
Means
1. 24, 36, 54
-24, 36, -54
2. 64, 16, 4
-64, 16, -4
3. 16 and -8
4.
𝟖
𝟑
-
𝟖
𝟑
5. 𝟒𝒙𝒚 𝟐
and 𝟖𝒙𝒚 𝟑

18. Assessment #1
HowWell DoYou
Know Me?
1. Geometric r = 3
2. Arithmetic d = 6
3. Neither
4. Geometric r = 1/5
5. Neither
6. Neither
7. Geometric r = ½
8. Arithmetic d = 2
Assessment #2
DoYou Remember
Me?
1. 𝒂 𝟓 = 80
2. 7
𝟏
𝟐
3. 9 and 27
4. 0
5. -5

19. Prepared and Submitted by:
ALMA E. BAJA
T-2
Approved:
JULITA M. VALDEZ
Principal II

20. I did great and fully
understood the lesson
51 - 67
I did good and partially
understood the lesson. I still
need to review.
21 - 50
I need to go back with the
lesson because there are
still areas I need to work
on. I will not give up.
1-20
I
My Score is
________