The document defines and provides examples of geometric sequences. A geometric sequence is a sequence where each term is found by multiplying the previous term by a common ratio. The general formula for a geometric sequence is an = a1rn-1, where a1 is the first term and r is the common ratio between terms. Examples show how to identify the common ratio r and use the general formula to determine the specific formula for different geometric sequences.

1. Geometric Sequences

2. Geometric Sequences
A sequence a1, a2 , a3 , … is an geometric sequence if
an = crn, i.e. it is defined by an exponential formula.

3. Geometric Sequences
A sequence a1, a2 , a3 , … is an geometric sequence if
an = crn, i.e. it is defined by an exponential formula.
Example A. The sequence of powers of 2
a1= 2, a2= 4, a3= 8, a4= 16, …
is an geometric sequence because an = 2n.

4. Geometric Sequences
A sequence a1, a2 , a3 , … is an geometric sequence if
an = crn, i.e. it is defined by an exponential formula.
Example A. The sequence of powers of 2
a1= 2, a2= 4, a3= 8, a4= 16, …
is an geometric sequence because an = 2n.
Fact: If a1, a2 , a3 , …an = c*rn is a geometric sequence, then
the ratio between any two consecutive terms is r.

5. Geometric Sequences
A sequence a1, a2 , a3 , … is an geometric sequence if
an = crn, i.e. it is defined by an exponential formula.
Example A. The sequence of powers of 2
a1= 2, a2= 4, a3= 8, a4= 16, …
is an geometric sequence because an = 2n.
Fact: If a1, a2 , a3 , …an = c*rn is a geometric sequence, then
the ratio between any two consecutive terms is r.
The converse of this fact is also true.

6. Geometric Sequences
A sequence a1, a2 , a3 , … is an geometric sequence if
an = crn, i.e. it is defined by an exponential formula.
Example A. The sequence of powers of 2
a1= 2, a2= 4, a3= 8, a4= 16, …
is an geometric sequence because an = 2n.
Fact: If a1, a2 , a3 , …an = c*rn is a geometric sequence, then
the ratio between any two consecutive terms is r.
The converse of this fact is also true. Below is the formula
that we used for working with geometric sequences.

7. Geometric Sequences
A sequence a1, a2 , a3 , … is an geometric sequence if
an = crn, i.e. it is defined by an exponential formula.
Example A. The sequence of powers of 2
a1= 2, a2= 4, a3= 8, a4= 16, …
is an geometric sequence because an = 2n.
Fact: If a1, a2 , a3 , …an = c*rn is a geometric sequence, then
the ratio between any two consecutive terms is r.
The converse of this fact is also true.
For example, from the sequence above we see that
16/8 = 8/4 = 4/2 = 2 = ratio r. Below is the formula that we
used for working with geometric sequences.

8. Geometric Sequences
A sequence a1, a2 , a3 , … is an geometric sequence if
an = crn, i.e. it is defined by an exponential formula.
Example A. The sequence of powers of 2
a1= 2, a2= 4, a3= 8, a4= 16, …
is an geometric sequence because an = 2n.
Fact: If a1, a2 , a3 , …an = c*rn is a geometric sequence, then
the ratio between any two consecutive terms is r.
The converse of this fact is also true. Below is the formula
that we used for working with geometric sequences.
For example, from the sequence above we see that
16/8 = 8/4 = 4/2 = 2 = ratio r
Theorem: If a1, a2 , a3 , …an is a sequence such that
an+1 / an = r for all n, then a1, a2, a3,… is an geometric
sequence and an = a1*rn-1.

9. Geometric Sequences
A sequence a1, a2 , a3 , … is an geometric sequence if
an = crn, i.e. it is defined by an exponential formula.
Example A. The sequence of powers of 2
a1= 2, a2= 4, a3= 8, a4= 16, …
is an geometric sequence because an = 2n.
Fact: If a1, a2 , a3 , …an = c*rn is a geometric sequence, then
the ratio between any two consecutive terms is r.
The converse of this fact is also true. Below is the formula
that we used for working with geometric sequences.
For example, from the sequence above we see that
16/8 = 8/4 = 4/2 = 2 = ratio r
Theorem: If a1, a2 , a3 , …an is a sequence such that
an+1 / an = r for all n, then a1, a2, a3,… is an geometric
sequence and an = a1*rn-1. This is the general formula for
geometric sequences.

10. Geometric Sequences
Given the description of a geometric sequence, we use the
general formula to find the specific formula for that sequence.

11. Geometric Sequences
Given the description of a geometric sequence, we use the
general formula to find the specific formula for that sequence.
Example A. The sequence 2, 6, 18, 54, … is an geometric
sequence because 6/2 = 18/6

12. Geometric Sequences
Given the description of a geometric sequence, we use the
general formula to find the specific formula for that sequence.
Example A. The sequence 2, 6, 18, 54, … is an geometric
sequence because 6/2 = 18/6 = 54/18

13. Geometric Sequences
Given the description of a geometric sequence, we use the
general formula to find the specific formula for that sequence.
Example A. The sequence 2, 6, 18, 54, … is an geometric
sequence because 6/2 = 18/6 = 54/18 = … = 3 = r.

14. Geometric Sequences
Given the description of a geometric sequence, we use the
general formula to find the specific formula for that sequence.
Example A. The sequence 2, 6, 18, 54, … is an geometric
sequence because 6/2 = 18/6 = 54/18 = … = 3 = r.
Since a1 = 2, set them in the general formula of the
geometric sequences an = a1r n – 1

15. Geometric Sequences
Given the description of a geometric sequence, we use the
general formula to find the specific formula for that sequence.
Example A. The sequence 2, 6, 18, 54, … is an geometric
sequence because 6/2 = 18/6 = 54/18 = … = 3 = r.
Since a1 = 2, set them in the general formula of the
geometric sequences an = a1r n – 1 , we get the specific
formula for this sequence an = 2*3(n – 1)

16. Geometric Sequences
Given the description of a geometric sequence, we use the
general formula to find the specific formula for that sequence.
Example A. The sequence 2, 6, 18, 54, … is an geometric
sequence because 6/2 = 18/6 = 54/18 = … = 3 = r.
Since a1 = 2, set them in the general formula of the
geometric sequences an = a1r n – 1 , we get the specific
formula for this sequence an = 2*3(n – 1)
If a1, a2 , a3 , …an is a geometric sequence such that the terms
alternate between positive and negative signs,
then r is negative.

17. Geometric Sequences
Given the description of a geometric sequence, we use the
general formula to find the specific formula for that sequence.
Example A. The sequence 2, 6, 18, 54, … is an geometric
sequence because 6/2 = 18/6 = 54/18 = … = 3 = r.
Since a1 = 2, set them in the general formula of the
geometric sequences an = a1r n – 1 , we get the specific
formula for this sequence an = 2*3(n – 1)
If a1, a2 , a3 , …an is a geometric sequence such that the terms
alternate between positive and negative signs,
then r is negative.
Example B. The sequence 2/3, -1, 3/2, -9/4, … is a geometric
sequence because
-1/(2/3) = (3/2) / (-1)

18. Geometric Sequences
Given the description of a geometric sequence, we use the
general formula to find the specific formula for that sequence.
Example A. The sequence 2, 6, 18, 54, … is an geometric
sequence because 6/2 = 18/6 = 54/18 = … = 3 = r.
Since a1 = 2, set them in the general formula of the
geometric sequences an = a1r n – 1 , we get the specific
formula for this sequence an = 2*3(n – 1)
If a1, a2 , a3 , …an is a geometric sequence such that the terms
alternate between positive and negative signs,
then r is negative.
Example B. The sequence 2/3, -1, 3/2, -9/4, … is a geometric
sequence because
-1/(2/3) = (3/2) / (-1) = (-9/4) /(3/2)

19. Geometric Sequences
Given the description of a geometric sequence, we use the
general formula to find the specific formula for that sequence.
Example A. The sequence 2, 6, 18, 54, … is an geometric
sequence because 6/2 = 18/6 = 54/18 = … = 3 = r.
Since a1 = 2, set them in the general formula of the
geometric sequences an = a1r n – 1 , we get the specific
formula for this sequence an = 2*3(n – 1)
If a1, a2 , a3 , …an is a geometric sequence such that the terms
alternate between positive and negative signs,
then r is negative.
Example B. The sequence 2/3, -1, 3/2, -9/4, … is a geometric
sequence because
-1/(2/3) = (3/2) / (-1) = (-9/4) /(3/2) = … = -3/2 = r.

20. Geometric Sequences
Given the description of a geometric sequence, we use the
general formula to find the specific formula for that sequence.
Example A. The sequence 2, 6, 18, 54, … is an geometric
sequence because 6/2 = 18/6 = 54/18 = … = 3 = r.
Since a1 = 2, set them in the general formula of the
geometric sequences an = a1r n – 1 , we get the specific
formula for this sequence an = 2*3(n – 1)
If a1, a2 , a3 , …an is a geometric sequence such that the terms
alternate between positive and negative signs,
then r is negative.
Example B. The sequence 2/3, -1, 3/2, -9/4, … is a geometric
sequence because
-1/(2/3) = (3/2) / (-1) = (-9/4) /(3/2) = … = -3/2 = r.
Since a1 = 2/3, the specific formula is
2 –3 n–1
an = 3 ( 2 )

21. Geometric Sequences
To use the geometric general formula to find the specific
formula, we need the first term a1 and the ratio r.

22. Geometric Sequences
To use the geometric general formula to find the specific
formula, we need the first term a1 and the ratio r.
Example C. Given that a1, a2 , a3 , …is a geometric sequence
with r = -2 and a5 = 12,

23. Geometric Sequences
To use the geometric general formula to find the specific
formula, we need the first term a1 and the ratio r.
Example C. Given that a1, a2 , a3 , …is a geometric sequence
with r = -2 and a5 = 12,
a. find a1

24. Geometric Sequences
To use the geometric general formula to find the specific
formula, we need the first term a1 and the ratio r.
Example C. Given that a1, a2 , a3 , …is a geometric sequence
with r = -2 and a5 = 12,
a. find a1
By that the general geometric formula
an = a1r n – 1, we get
a5 = a1(-2)(5 – 1)

25. Geometric Sequences
To use the geometric general formula to find the specific
formula, we need the first term a1 and the ratio r.
Example C. Given that a1, a2 , a3 , …is a geometric sequence
with r = -2 and a5 = 12,
a. find a1
By that the general geometric formula
an = a1r n – 1, we get
a5 = a1(-2)(5 – 1) = 12

26. Geometric Sequences
To use the geometric general formula to find the specific
formula, we need the first term a1 and the ratio r.
Example C. Given that a1, a2 , a3 , …is a geometric sequence
with r = -2 and a5 = 12,
a. find a1
By that the general geometric formula
an = a1r n – 1, we get
a5 = a1(-2)(5 – 1) = 12
a1(-2)4 = 12

27. Geometric Sequences
To use the geometric general formula to find the specific
formula, we need the first term a1 and the ratio r.
Example C. Given that a1, a2 , a3 , …is a geometric sequence
with r = -2 and a5 = 12,
a. find a1
By that the general geometric formula
an = a1r n – 1, we get
a5 = a1(-2)(5 – 1) = 12
a1(-2)4 = 12
16a1 = 12

28. Geometric Sequences
To use the geometric general formula to find the specific
formula, we need the first term a1 and the ratio r.
Example C. Given that a1, a2 , a3 , …is a geometric sequence
with r = -2 and a5 = 12,
a. find a1
By that the general geometric formula
an = a1r n – 1, we get
a5 = a1(-2)(5 – 1) = 12
a1(-2)4 = 12
16a1 = 12
a1 = 12/16 = ¾

29. Geometric Sequences
To use the geometric general formula to find the specific
formula, we need the first term a1 and the ratio r.
Example C. Given that a1, a2 , a3 , …is a geometric sequence
with r = -2 and a5 = 12,
a. find a1
By that the general geometric formula
an = a1r n – 1, we get
a5 = a1(-2)(5 – 1) = 12
a1(-2)4 = 12
16a1 = 12
a1 = 12/16 = ¾
b. find the specific equation.

30. Geometric Sequences
To use the geometric general formula to find the specific
formula, we need the first term a1 and the ratio r.
Example C. Given that a1, a2 , a3 , …is a geometric sequence
with r = -2 and a5 = 12,
a. find a1
By that the general geometric formula
an = a1r n – 1, we get
a5 = a1(-2)(5 – 1) = 12
a1(-2)4 = 12
16a1 = 12
a1 = 12/16 = ¾
b. find the specific equation.
Set a1 = ¾ and r = -2 into the general formula an = a1rn – 1 ,

31. Geometric Sequences
To use the geometric general formula to find the specific
formula, we need the first term a1 and the ratio r.
Example C. Given that a1, a2 , a3 , …is a geometric sequence
with r = -2 and a5 = 12,
a. find a1
By that the general geometric formula
an = a1r n – 1, we get
a5 = a1(-2)(5 – 1) = 12
a1(-2)4 = 12
16a1 = 12
a1 = 12/16 = ¾
b. find the specific equation.
Set a1 = ¾ and r = -2 into the general formula an = a1rn – 1 ,
we get the specific formula of this sequence
an= 3 (-2)n–1
4

32. Geometric Sequences
C. Find a9.

33. Geometric Sequences
C. Find a9.
Since an= 3 (-2)n–1,
4

34. Geometric Sequences
C. Find a9.
Since an= 3 (-2)n–1,
4
set n = 9, we get
a9= 3 (-2)9–1
4

35. Geometric Sequences
C. Find a9.
Since an= 3 (-2)n–1,
4
set n = 9, we get
a9= 3 (-2)9–1
4
3
a9 = 4 (-2)8

36. Geometric Sequences
C. Find a9.
Since an= 3 (-2)n–1,
4
set n = 9, we get
a9= 3 (-2)9–1
4
3 3
a9 = 4 (-2)8 = 4 (256) = 192

37. Geometric Sequences
C. Find a9.
Since an= 3 (-2)n–1,
4
set n = 9, we get
a9= 3 (-2)9–1
4
3 3
a9 = 4 (-2)8 = 4 (256) = 192
Example D. Given that a1, a2 , a3 , …is an geometric
sequence with a3 = -2 and a6 = 54,

38. Geometric Sequences
C. Find a9.
Since an= 3 (-2)n–1,
4
set n = 9, we get
a9= 3 (-2)9–1
4
3 3
a9 = 4 (-2)8 = 4 (256) = 192
Example D. Given that a1, a2 , a3 , …is an geometric
sequence with a3 = -2 and a6 = 54,
a. find r and a1

39. Geometric Sequences
C. Find a9.
Since an= 3 (-2)n–1,
4
set n = 9, we get
a9= 3 (-2)9–1
4
3 3
a9 = 4 (-2)8 = 4 (256) = 192
Example D. Given that a1, a2 , a3 , …is an geometric
sequence with a3 = -2 and a6 = 54,
a. find r and a1
Given that the general geometric formula an = a1rn – 1,

40. Geometric Sequences
C. Find a9.
Since an= 3 (-2)n–1,
4
set n = 9, we get
a9= 3 (-2)9–1
4
3 3
a9 = 4 (-2)8 = 4 (256) = 192
Example D. Given that a1, a2 , a3 , …is an geometric
sequence with a3 = -2 and a6 = 54,
a. find r and a1
Given that the general geometric formula an = a1rn – 1,
we have
a3 = -2 = a1r3–1

41. Geometric Sequences
C. Find a9.
Since an= 3 (-2)n–1,
4
set n = 9, we get
a9= 3 (-2)9–1
4
3 3
a9 = 4 (-2)8 = 4 (256) = 192
Example D. Given that a1, a2 , a3 , …is an geometric
sequence with a3 = -2 and a6 = 54,
a. find r and a1
Given that the general geometric formula an = a1rn – 1,
we have
a3 = -2 = a1r3–1 and a6 = 54 = a1r6–1

42. Geometric Sequences
C. Find a9.
Since an= 3 (-2)n–1,
4
set n = 9, we get
a9= 3 (-2)9–1
4
3 3
a9 = 4 (-2)8 = 4 (256) = 192
Example D. Given that a1, a2 , a3 , …is an geometric
sequence with a3 = -2 and a6 = 54,
a. find r and a1
Given that the general geometric formula an = a1rn – 1,
we have
a3 = -2 = a1r3–1 and a6 = 54 = a1r6–1
-2 = a1r2

43. Geometric Sequences
C. Find a9.
Since an= 3 (-2)n–1,
4
set n = 9, we get
a9= 3 (-2)9–1
4
3 3
a9 = 4 (-2)8 = 4 (256) = 192
Example D. Given that a1, a2 , a3 , …is an geometric
sequence with a3 = -2 and a6 = 54,
a. find r and a1
Given that the general geometric formula an = a1rn – 1,
we have
a3 = -2 = a1r3–1 and a6 = 54 = a1r6–1
-2 = a1r2 54 = a 1r5

44. Geometric Sequences
C. Find a9.
Since an= 3 (-2)n–1,
4
set n = 9, we get
a9= 3 (-2)9–1
4
3 3
a9 = 4 (-2)8 = 4 (256) = 192
Example D. Given that a1, a2 , a3 , …is an geometric
sequence with a3 = -2 and a6 = 54,
a. find r and a1
Given that the general geometric formula an = a1rn – 1,
we have
a3 = -2 = a1r3–1 and a6 = 54 = a1r6–1
-2 = a1r2 54 = a 1r5
Divide these equations:

45. Geometric Sequences
54 a1r5
= a r2
-2 1

46. Geometric Sequences
-27 54 = a1r
5
-2 a1r2

47. Geometric Sequences
-27 54 = a1r
5
-2 a1r2

48. Geometric Sequences
3 = 5-2
-27 54 = a1r
5
-2 a1r2

49. Geometric Sequences
3 = 5-2
-27 54 = a1r
5
-2 a1r2
-27 = r3

50. Geometric Sequences
3 = 5-2
-27 54 = a1r
5
-2 a1r2
-27 = r3
-3 = r

51. Geometric Sequences
5 3 = 5-2
-27 54 = a1r
-2 a1r2
-27 = r3
-3 = r
Put r = -3 into the equation -2 = a1r2

52. Geometric Sequences
5 3 = 5-2
-27 54 = a1r
-2 a1r2
-27 = r3
-3 = r
Put r = -3 into the equation -2 = a1r2
Hence -2 = a1(-3)2

53. Geometric Sequences
5 3 = 5-2
-27 54 = a1r
-2 a1r2
-27 = r3
-3 = r
Put r = -3 into the equation -2 = a1r2
Hence -2 = a1(-3)2
-2 = a19

54. Geometric Sequences
5 3 = 5-2
-27 54 = a1r
-2 a1r2
-27 = r3
-3 = r
Put r = -3 into the equation -2 = a1r2
Hence -2 = a1(-3)2
-2 = a19
-2/9 = a1

55. Geometric Sequences
5 3 = 5-2
-27 54 = a1r
-2 a1r2
-27 = r3
-3 = r
Put r = -3 into the equation -2 = a1r2
Hence -2 = a1(-3)2
-2 = a19
-2/9 = a1
b. Find the specific formula and a2

56. Geometric Sequences
5 3 = 5-2
-27 54 = a1r
-2 a1r2
-27 = r3
-3 = r
Put r = -3 into the equation -2 = a1r2
Hence -2 = a1(-3)2
-2 = a19
-2/9 = a1
b. Find the specific formula and a2
Use the general geometric formula an = a1rn – 1,
set a1 = -2/9, and r = -3

57. Geometric Sequences
5 3 = 5-2
-27 54 = a1r
-2 a1r2
-27 = r3
-3 = r
Put r = -3 into the equation -2 = a1r2
Hence -2 = a1(-3)2
-2 = a19
-2/9 = a1
b. Find the specific formula and a2
Use the general geometric formula an = a1rn – 1,
set a1 = -2/9, and r = -3 we have the specific formula
an = -2 (-3)n–1
9

58. Geometric Sequences
5 3 = 5-2
-27 54 = a1r
-2 a1r2
-27 = r3
-3 = r
Put r = -3 into the equation -2 = a1r2
Hence -2 = a1(-3)2
-2 = a19
-2/9 = a1
b. Find the specific formula and a2
Use the general geometric formula an = a1rn – 1,
set a1 = -2/9, and r = -3 we have the specific formula
an = -2 (-3)n–1
9
To find a2, set n = 2, we get
a2 = -2 (-3)
2–1
9

59. Geometric Sequences
5 3 = 5-2
-27 54 = a1r
-2 a1r2
-27 = r3
-3 = r
Put r = -3 into the equation -2 = a1r2
Hence -2 = a1(-3)2
-2 = a19
-2/9 = a1
b. Find the specific formula and a2
Use the general geometric formula an = a1rn – 1,
set a1 = -2/9, and r = -3 we have the specific formula
an = -2 (-3)n–1
9
To find a2, set n = 2, we get
a2 = -2 (-3) = -2 (-3)
2–1
9 9

60. Geometric Sequences
5 3 = 5-2
-27 54 = a1r
-2 a1r2
-27 = r3
-3 = r
Put r = -3 into the equation -2 = a1r2
Hence -2 = a1(-3)2
-2 = a19
-2/9 = a1
b. Find the specific formula and a2
Use the general geometric formula an = a1rn – 1,
set a1 = -2/9, and r = -3 we have the specific formula
an = -2 (-3)n–1
9
To find a2, set n = 2, we get
a2 = -2 (-3) = -2 3 (-3)
2–1
9 9

61. Geometric Sequences
5 3 = 5-2
-27 54 = a1r
-2 a1r2
-27 = r3
-3 = r
Put r = -3 into the equation -2 = a1r2
Hence -2 = a1(-3)2
-2 = a19
-2/9 = a1
b. Find the specific formula and a2
Use the general geometric formula an = a1rn – 1,
set a1 = -2/9, and r = -3 we have the specific formula
an = -2 (-3)n–1
9
To find a2, set n = 2, we get
a2 = -2 (-3) = -2 3 (-3) = 2
2–1
9 9 3

62. Geometric Sequences
Sum of geometric sequences

63. Geometric Sequences
Sum of geometric sequences
We observe the algebraic patterns:
(1 – r)(1 + r) = 1 – r2

64. Geometric Sequences
Sum of geometric sequences
We observe the algebraic patterns:
(1 – r)(1 + r) = 1 – r2
(1 – r)(1 + r + r2) = 1 – r3

65. Geometric Sequences
Sum of geometric sequences
We observe the algebraic patterns:
(1 – r)(1 + r) = 1 – r2
(1 – r)(1 + r + r2) = 1 – r3
(1 – r)(1 + r + r2 + r3) = 1 – r4

66. Geometric Sequences
Sum of geometric sequences
We observe the algebraic patterns:
(1 – r)(1 + r) = 1 – r2
(1 – r)(1 + r + r2) = 1 – r3
(1 – r)(1 + r + r2 + r3) = 1 – r4
(1 – r)(1 + r + r2 + r3 + r4) = 1 – r5

67. Geometric Sequences
Sum of geometric sequences
We observe the algebraic patterns:
(1 – r)(1 + r) = 1 – r2
(1 – r)(1 + r + r2) = 1 – r3
(1 – r)(1 + r + r2 + r3) = 1 – r4
(1 – r)(1 + r + r2 + r3 + r4) = 1 – r5
...
…
(1 – r)(1 + r + r2 … + rn-1) = 1 – rn

68. Geometric Sequences
Sum of geometric sequences
We observe the algebraic patterns:
(1 – r)(1 + r) = 1 – r2
(1 – r)(1 + r + r2) = 1 – r3
(1 – r)(1 + r + r2 + r3) = 1 – r4
(1 – r)(1 + r + r2 + r3 + r4) = 1 – r5
...
…
(1 – r)(1 + r + r2 … + rn-1) = 1 – rn
1 – rn
Hence 1 + r + r + … + r = 1 – r
2 n-1

69. Geometric Sequences
Sum of geometric sequences
We observe the algebraic patterns:
(1 – r)(1 + r) = 1 – r2
(1 – r)(1 + r + r2) = 1 – r3
(1 – r)(1 + r + r2 + r3) = 1 – r4
(1 – r)(1 + r + r2 + r3 + r4) = 1 – r5
...
…
(1 – r)(1 + r + r2 … + rn-1) = 1 – rn
1 – rn
Hence 1 + r + r + … + r = 1 – r
2 n-1
Therefore a1 + a1r + a1r2 + … +a1rn-1

70. Geometric Sequences
Sum of geometric sequences
We observe the algebraic patterns:
(1 – r)(1 + r) = 1 – r2
(1 – r)(1 + r + r2) = 1 – r3
(1 – r)(1 + r + r2 + r3) = 1 – r4
(1 – r)(1 + r + r2 + r3 + r4) = 1 – r5
...
…
(1 – r)(1 + r + r2 … + rn-1) = 1 – rn
1 – rn
Hence 1 + r + r + … + r = 1 – r
2 n-1
Therefore a1 + a1r + a1r2 + … +a1rn-1
n terms

71. Geometric Sequences
Sum of geometric sequences
We observe the algebraic patterns:
(1 – r)(1 + r) = 1 – r2
(1 – r)(1 + r + r2) = 1 – r3
(1 – r)(1 + r + r2 + r3) = 1 – r4
(1 – r)(1 + r + r2 + r3 + r4) = 1 – r5
...
…
(1 – r)(1 + r + r2 … + rn-1) = 1 – rn
1 – rn
Hence 1 + r + r + … + r = 1 – r
2 n-1
Therefore a1 + a1r + a1r2 + … +a1rn-1
n terms
= a1(1 + r + r2 + … + r n-1)

72. Geometric Sequences
Sum of geometric sequences
We observe the algebraic patterns:
(1 – r)(1 + r) = 1 – r2
(1 – r)(1 + r + r2) = 1 – r3
(1 – r)(1 + r + r2 + r3) = 1 – r4
(1 – r)(1 + r + r2 + r3 + r4) = 1 – r5
...
…
(1 – r)(1 + r + r2 … + rn-1) = 1 – rn
1 – rn
Hence 1 + r + r + … + r = 1 – r
2 n-1
Therefore a1 + a1r + a1r2 + … +a1rn-1
n terms
a1 1 – r
n
= a1(1 + r + r + … + r ) =
2 n-1
1–r

73. Geometric Sequences
Formula for the Sum Geometric Sequences
1 – rn
a1 + a1r + a1r + … +a1r = a1 1 – r
2 n-1

74. Geometric Sequences
Formula for the Sum Geometric Sequences
1 – rn
a1 + a1r + a1r + … +a1r = a1 1 – r
2 n-1
Example E. Find the geometric sum :
2/3 + (-1) + 3/2 + … + (-81/16)

75. Geometric Sequences
Formula for the Sum Geometric Sequences
1 – rn
a1 + a1r + a1r + … +a1r = a1 1 – r
2 n-1
Example E. Find the geometric sum :
2/3 + (-1) + 3/2 + … + (-81/16)
We have a1 = 2/3 and r = -3/2,

76. Geometric Sequences
Formula for the Sum Geometric Sequences
1 – rn
a1 + a1r + a1r + … +a1r = a1 1 – r
2 n-1
Example E. Find the geometric sum :
2/3 + (-1) + 3/2 + … + (-81/16)
We have a1 = 2/3 and r = -3/2, and an = -81/16.

77. Geometric Sequences
Formula for the Sum Geometric Sequences
1 – rn
a1 + a1r + a1r + … +a1r = a1 1 – r
2 n-1
Example E. Find the geometric sum :
2/3 + (-1) + 3/2 + … + (-81/16)
We have a1 = 2/3 and r = -3/2, and an = -81/16. We need the
number of terms.

78. Geometric Sequences
Formula for the Sum Geometric Sequences
1 – rn
a1 + a1r + a1r + … +a1r = a1 1 – r
2 n-1
Example E. Find the geometric sum :
2/3 + (-1) + 3/2 + … + (-81/16)
We have a1 = 2/3 and r = -3/2, and an = -81/16. We need the
number of terms. Put a1 and r in the general formula we get
the specific formula
an= 2 ( - 3 ) n-1
3 2

79. Geometric Sequences
Formula for the Sum Geometric Sequences
1 – rn
a1 + a1r + a1r + … +a1r = a1 1 – r
2 n-1
Example E. Find the geometric sum :
2/3 + (-1) + 3/2 + … + (-81/16)
We have a1 = 2/3 and r = -3/2, and an = -81/16. We need the
number of terms. Put a1 and r in the general formula we get
the specific formula
an= 2 ( - 3 ) n-1
3 2
To find n, set an = -81
16

80. Geometric Sequences
Formula for the Sum Geometric Sequences
1 – rn
a1 + a1r + a1r + … +a1r = a1 1 – r
2 n-1
Example E. Find the geometric sum :
2/3 + (-1) + 3/2 + … + (-81/16)
We have a1 = 2/3 and r = -3/2, and an = -81/16. We need the
number of terms. Put a1 and r in the general formula we get
the specific formula
an= 2 ( - 3 ) n-1
3 2
To find n, set an = -81 = 2 ( - 3 ) n – 1
16 3 2

81. Geometric Sequences
Formula for the Sum Geometric Sequences
1 – rn
a1 + a1r + a1r + … +a1r = a1 1 – r
2 n-1
Example E. Find the geometric sum :
2/3 + (-1) + 3/2 + … + (-81/16)
We have a1 = 2/3 and r = -3/2, and an = -81/16. We need the
number of terms. Put a1 and r in the general formula we get
the specific formula
an= 2 ( - 3 ) n-1
3 2
To find n, set an = -81 = 2 ( - 3 ) n – 1
16 3 2
-243 = ( - 3 ) n – 1
32 2

82. Geometric Sequences
Formula for the Sum Geometric Sequences
1 – rn
a1 + a1r + a1r + … +a1r = a1 1 – r
2 n-1
Example E. Find the geometric sum :
2/3 + (-1) + 3/2 + … + (-81/16)
We have a1 = 2/3 and r = -3/2, and an = -81/16. We need the
number of terms. Put a1 and r in the general formula we get
the specific formula
an= 2 ( - 3 ) n-1
3 2
To find n, set an = -81 = 2 ( - 3 ) n – 1
16 3 2
-243 = ( - 3 ) n – 1
32 2
Compare the denominators we see that 32 = 2n – 1.

83. Geometric Sequences
Formula for the Sum Geometric Sequences
1 – rn
a1 + a1r + a1r + … +a1r = a1 1 – r
2 n-1
Example E. Find the geometric sum :
2/3 + (-1) + 3/2 + … + (-81/16)
We have a1 = 2/3 and r = -3/2, and an = -81/16. We need the
number of terms. Put a1 and r in the general formula we get
the specific formula
an= 2 ( - 3 ) n-1
3 2
To find n, set an = -81 = 2 ( - 3 ) n – 1
16 3 2
-243 = ( - 3 ) n – 1
32 2
Compare the denominators we see that 32 = 2n – 1.
Since 32 = 25 = 2n – 1

84. Geometric Sequences
Formula for the Sum Geometric Sequences
1 – rn
a1 + a1r + a1r + … +a1r = a1 1 – r
2 n-1
Example E. Find the geometric sum :
2/3 + (-1) + 3/2 + … + (-81/16)
We have a1 = 2/3 and r = -3/2, and an = -81/16. We need the
number of terms. Put a1 and r in the general formula we get
the specific formula
an= 2 ( - 3 ) n-1
3 2
To find n, set an = -81 = 2 ( - 3 ) n – 1
16 3 2
-243 = ( - 3 ) n – 1
32 2
Compare the denominators we see that 32 = 2n – 1.
Since 32 = 25 = 2n – 1
n–1=5

85. Geometric Sequences
Formula for the Sum Geometric Sequences
1 – rn
a1 + a1r + a1r + … +a1r = a1 1 – r
2 n-1
Example E. Find the geometric sum :
2/3 + (-1) + 3/2 + … + (-81/16)
We have a1 = 2/3 and r = -3/2, and an = -81/16. We need the
number of terms. Put a1 and r in the general formula we get
the specific formula
an= 2 ( - 3 ) n-1
3 2
To find n, set an = -81 = 2 ( - 3 ) n – 1
16 3 2
-243 = ( - 3 ) n – 1
32 2
Compare the denominators we see that 32 = 2n – 1.
Since 32 = 25 = 2n – 1
n–1=5
n=6

86. Geometric Sequences
Therefore there are 6 terms in the sum,
2/3 + (-1) + 3/2 + … + (-81/16)

87. Geometric Sequences
Therefore there are 6 terms in the sum,
2/3 + (-1) + 3/2 + … + (-81/16)
1 – rn
Set a1 = 2/3, r = -3/2 and n = 6 in the formula S = a1 1 – r

88. Geometric Sequences
Therefore there are 6 terms in the sum,
2/3 + (-1) + 3/2 + … + (-81/16)
1 – rn
Set a1 = 2/3, r = -3/2 and n = 6 in the formula S = a1 1 – r
we get the sum S
2 1 – (-3/2)6
S = 3 1 – (-3/2)

89. Geometric Sequences
Therefore there are 6 terms in the sum,
2/3 + (-1) + 3/2 + … + (-81/16)
1 – rn
Set a1 = 2/3, r = -3/2 and n = 6 in the formula S = a1 1 – r
we get the sum S
2 1 – (-3/2)6
S = 3 1 – (-3/2)
= 2 1 – (729/64)
3 1 + (3/2)

90. Geometric Sequences
Therefore there are 6 terms in the sum,
2/3 + (-1) + 3/2 + … + (-81/16)
1 – rn
Set a1 = 2/3, r = -3/2 and n = 6 in the formula S = a1 1 – r
we get the sum S
2 1 – (-3/2)6
S = 3 1 – (-3/2)
= 2 1 – (729/64)
3 1 + (3/2)
= 2 -665/64
3 5/2

91. Geometric Sequences
Therefore there are 6 terms in the sum,
2/3 + (-1) + 3/2 + … + (-81/16)
1 – rn
Set a1 = 2/3, r = -3/2 and n = 6 in the formula S = a1 1 – r
we get the sum S
2 1 – (-3/2)6
S = 3 1 – (-3/2)
= 2 1 – (729/64)
3 1 + (3/2)
= 2 -665/64
3 5/2
= -133
48

92. Geometric Sequences
Therefore there are 6 terms in the sum,
2/3 + (-1) + 3/2 + … + (-81/16)
1 – rn
Set a1 = 2/3, r = -3/2 and n = 6 in the formula S = a1 1 – r
we get the sum S
2 1 – (-3/2)6
S = 3 1 – (-3/2)
= 2 1 – (729/64)
3 1 + (3/2)
= 2 -665/64
3 5/2
= -133
48

93. Geometric Sequences
Therefore there are 6 terms in the sum,
2/3 + (-1) + 3/2 + … + (-81/16)
1 – rn
Set a1 = 2/3, r = -3/2 and n = 6 in the formula S = a1 1 – r
we get the sum S
2 1 – (-3/2)6
S = 3 1 – (-3/2)
= 2 1 – (729/64)
3 1 + (3/2)
= 2 -665/64
3 5/2
= -133
48

94. Geometric Sequences
HW.
Given that a1, a2 , a3 , …is a geometric sequence find a1, r,
and the specific formula for the an.
1. a2 = 15, a5 = 405
2. a4 = –5/2, a8 = –40
2. a3 = 3/4, a6 = –2/9
Sum the following geometric sequences.
1. 3 + 6 + 12 + .. + 3072
1. –2 + 6 –18 + .. + 486
1. 6 – 3 + 3/2 – .. + 3/512