Contents: 1. Geometric Sequence 2. Geometric Means 3. Geometric Series with activities Feel free to send me a message at regie.naungayan@deped.gov.ph for corrections or other suggestions

1. GEOMETRIC
SEQUENCE & SERIES
MR. REGIE R. NAUNGAYAN
TEACHER III
DEPED BANAYOYO NATIONAL HIGH SCHOOL

2. OBJECTIVES
After going through this module, the learner should be
able to:
a. determine a geometric sequence;
b. identify the common ratio of a geometric sequence;
c. find the missing term of a geometric sequence; and,
d. determine whether a sequence is geometric or arithmetic.
e. insert geometric means in a sequence
f. solve the sum of a geometric sequence.

3. WHAT IS A GEOMETRIC SEQUENCE?
A geometric sequence is a
sequence obtained by
multiplying a common ratio to
the preceding terms in order to
obtain the succeeding terms.

4. HOW TO DETERMINE THE COMMON RATIO?
The common ratio is obtained by
dividing a term by the term preceding it.
𝑎2
𝑎1
,
𝑎3
𝑎2
,
𝑎4
𝑎3
, …
࢙࢛ࢉࢉࢋࢋࢊ࢏࢔ࢍ ࢚ࢋ࢘࢓
࢖࢘ࢋࢉࢋࢊ࢏࢔ࢍ ࢚ࢋ࢘࢓

5. EXAMPLES
1, 2, 4, 8, 16, …
3, -6, 12,-24, …
3, 1,
1
3
,
1
9
,
1
27
… 1
3

6. ACTIVITY 1: 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. 10/3, 10/6, 10/9, 10/15, …
Geometric
Not Geometric
Geometric
Geometric
Not Geometric

7. Identify which of the following is a geometric sequence. If observed as a
geometric sequence, determine r.
1. 1, 2, 4, 12, 16, … 6. 1, 3, 9, 27, 81, . . .
2. -2, 4, -8, 16, -32, … 7. 27, 9, 3, 1, 1/3, …
3. 10, 20, 30, 40, 50… 8. 3, 12, 48, 192, 768, . . .
4. 5, 10, 20, 40, 80, … 9. 2, 26, 338, 4 394, …
5. 4, 12, 36, 108, 324 … 10. 5x2, 5x5, 5x8, 5x11, 5x14,
…
X
r = -2
X
r = 2
r = 3
r = 3
r = 1/3
r = 4
r = 13
r = 𝒙𝟑
ACTIVITY 2: GEOMETRIC BA ‘YAN?

8. Arithmetic
Sequence
Geometric
Sequence

9. ACTIVITY 3: SHADE THAT BOX
WORKSHEET
Shade the box with
blue if the sequence
inside it is geometric,
but color it red if it
contains an arithmetic
sequence. Leave the
box uncolored if
neither a geometric nor
arithmetic.

10. ACTIVITY 3: SHADE THAT BOX (ANSWER)

11. FINDING NTH
GEOMETRIC SEQUENCE

12. an = a1rn-1
FORMULA

13. EXAMPLE
Find the 9th term of the sequence
1, 3, 9, 27, …
a1 = 1
r = 3
n = 9
an = a1rn-1
a9 = (1)(3)9-1
a9 = (1)(3)8
a9 = (1)(6561)
a9 = 6561

14. EXAMPLE
Find the 10th term of the sequence
-2, 8, -32, 128, …
a1 = -2
r = -4
n = 10
an = a1rn-1
a10 = (-2)(-4)10-1
a10 = (-2)(-4)9
a10 = (-2)(-262,144)
a10 = 524,288

15. EXAMPLE
What is the 5th term of the geometric
sequence whose a1 =
1
5
, and r = 2?
an = a1rn-1
a5 =
1
5
(2)5-1
a5 =
1
5
(2)4
a5 =
1
5
(16)
a5 =
16
5

16. TRY THESE!
1. What is the 5th term of the geometric
sequence whose a1 = 100, and r = -
1
2
?
2. Find a7 of the geometric sequence with
a1 = −7, and r = 3?
a7 = -5,103
a5 =
𝟐𝟓
𝟒

17. GEOMETRIC
MEAN

18. ACTIVITY 4: WHAT’S THAT TERM?
Find the specified term in each of the following geometric sequence.
1. 3, 9, 27, … a6
2. 1, -2, 4, -8, … a7
3. 12, a2, 3
4. 2, a2, a3, 54
a6 = 729
a7 = 64
a2 = 6
a2 = 6, a2 = 18

19. DEFINITION
The given terms are the first and
last terms. These terms are called
the extremes, and the term/s in
between the extremes are called
geometric mean/s.

20. DEFINITION
2, 4, 8, 16, 32

21. FORMULA
To insert terms, let us identify
first the common ratio by using the
formula:
𝒓 =
𝒏−𝟏 𝒂𝒏
𝒂𝟏

22. EXAMPLE
Find the geometric mean of 4 and 484.
𝒓 =
𝒏−𝟏 𝒂𝒏
𝒂𝟏
𝒓 =
𝟑−𝟏 𝟒𝟖𝟒
𝟒
𝒓 =
𝟐
𝟏𝟐𝟏
𝒓 = ± 11
4, ___, 484
a1 an
n = 3

23. EXAMPLE Find the geometric mean of 4 and 484.
 If 𝑟 = 11, then the sequence is 4, 44, 484,
which means that the geometric mean is
44.
 If 𝑟 = -11, then the sequence is 4, -44, 484,
which means that the geometric mean is -
44.
𝑟 = ±11 denotes two possible answers

24. EXAMPLE
Insert two geometric means between 6
and 750.
𝒓 =
𝒏−𝟏 𝒂𝒏
𝒂𝟏
𝒓 =
𝟒−𝟏 𝟕𝟓𝟎
𝟔
𝒓 =
𝟑
𝟏𝟐𝟓
𝒓 = 5
6, ___, ___,
750
a1 an
n = 4
If r = 5, then the geometric
means are 30 and 150.
30 150

25. EXAMPLE
Insert two geometric means between 5 184
and 3.
𝒓 =
𝒏−𝟏 𝒂𝒏
𝒂𝟏
𝒓 =
𝟒−𝟏 𝟑
𝟓𝟏𝟖𝟒
𝒓 =
𝟑 𝟏
𝟏𝟕𝟐𝟖
𝒓 =
𝟏
𝟏𝟐
5184, ___, ___, 3
a1 an
n = 4
43
2
36

26. YOUR TURN!
1. Insert three geometric means
between 7 and 16 807.
2. Insert four geometric means
between 5 and 38 880.
30, 180, 1080, 6480
49, 343, 2401 or -49, 343, -2401

27. GEOMETRIC
SERIES

28. FORMULA
Sn =
a1(1 −rn)
1 −r

29. EXAMPLE
Find the sum of the first 6 terms of a sequence
with a1 = 4, r = 5.
a1 = 4
r = 5
n = 6
Sn =
a1(1 −rn)
1 −r
S6 =
4 (1 −(5)6)
1 − 5
S6 =
4 (1 − 15 625)
−4
S6 =
4 (−15 624)
−4
S6 =
−62 496
−4
S6 = 15 624

30. EXAMPLE
Find the sum of the first 12 terms of the geometric
sequence 3, -9, 27, -81, 243, …
a1 = 3
r = -3
n = 12
Sn =
a1(1 −rn)
1 −r
S12 =
3 (1 −(−3)12)
1 −(−3)
S12 =
3 (1 −531 441)
1+3
S12 =
3 (−531 440)
4
S12 =
−1 594 320
4
S12 = -398 580

31. EXAMPLE
If a1 = 318, r =
1
2
, what is the sum of the first 7
terms?
a1 =
318
r =
1
2
n = 7
Sn =
a1(1 −rn)
1 −r
S7 =
318 1 −
1
2
7
1 −
1
2
S7 =
318 1 −
1
128
1
2
S7 =
318
127
128
1
2
S7 = 318
127
128
2
1
S7 =
20 193
32

32. Find the sum of the indicated series.
1. a1 = 4, r = 2, n = 20
2. a1 = -5, r = 3, n = 9
3. a1 = 216, r =
1
2
, n = 5
4. -6, 24, -96, 384, … n = 7
5. 1.6, 0.08, 0.004, … n = 5
ACTIVITY 5: SUM IT UP!