This document discusses geometric sequences and series. It defines a geometric sequence as a sequence whose consecutive terms have a common ratio. The nth term of a geometric sequence can be found using the formula an = a1rn-1, where a1 is the first term and r is the common ratio. It also provides the formula to find the sum of a finite geometric series, which is S = a1(1-rn)/(1-r), where S is the sum, a1 is the first term, r is the common ratio, and n is the number of terms. Several examples are provided to demonstrate finding terms, sums, and determining if a sequence is geometric.

1. Geometric Series and
Geometric Sequences
Lesson 6
Prepared by:
JERREY G. LUNGAY

2. Essential Question: What is a sequence
and how do I find its terms and sums?
How do I find the sum & terms of
geometric sequences and series?

3. Geometric Sequences
Geometric Sequence– a sequence whose
consecutive terms have a common ratio.

4. A sequence is geometric if the ratios of
consecutive terms are the same.
Geometric Sequence
3
2 4
1 2 3
.....
a
a a
r
a a a
   
The number r is the common ratio.

5. 2, 4, 8, 16, …, formula?, …
Ex. 1
12, 36, 108, 324, …, formula?, …
1, 4, 9, 16, …, formula? , …
Are these geometric?
1 1 1 1
, , , ,..., ?,...
3 9 27 61
formula
 
Yes 2n
Yes
4(3)n
No n2
No
(-1)n /3

6. Finding the nth term of a Geometric
Sequence
an = a1rn – 1
r
a
a
 2
1

7. Ex. 2b
Write the first five terms of the
geometric sequence whose first
term is a1 = 9 and r = (1/3).
9 3 1
1
3
1
9
, , , ,

8. Ex. 3
Find the 15th term of the geometric
sequence whose first term is 20 and
whose common ratio is 1.05
an = a1rn – 1
a15 = (20)(1.05)15 – 1
a15 = 39.599

9. Ex. 4 Find a formula for the nth term.
What is the 9th term?
5, 15, 45, …
an = 5(3)n – 1
an = 5(3)n – 1
a9 = 5(3)8
a9 = 32805
an = a1rn – 1

10. s
a r
r
n
n



1 1
1
( )
sum of a finite geometric series

11. Ex. 6 Find the sum of the first 12 terms of the
series 4(0.3)n
= 4(0.3)1 + 4(0.3)2 + 4(0.3)3 + … + 4(0.3)12
r
r
a
a
S
n
n



1
1
1
3
.
0
1
)
3
.
0
(
2
.
1
2
.
1 12
12



S = 1.714

12. Ex. 7 Find the sum of the first 5 terms of the
series 5/3 + 5 + 15 + …
r = 5/(5/3) = 3
r
r
a
a
S
n
n



1
1
1
3
1
)
3
(
3
5
3
5 5
5



S
= 605/3

13. 1, 4, 7, 10, 13, …. Infinite Arithmetic No Sum
3, 7, 11, …, 51 Finite Arithmetic  
n 1 n
n
S a a
2
 
1, 2, 4, …, 64 Finite Geometric
 
n
1
n
a r 1
S
r 1



1, 2, 4, 8, … Infinite Geometric
r > 1
r < -1
No Sum
1 1 1
3,1, , , ...
3 9 27
Infinite Geometric
-1 < r < 1
1
a
S
1 r



14. Find the sum, if possible:
1 1 1
1 ...
2 4 8
   
1 1
1
2 4
r
1
1 2
2
   1 r 1 Yes
    
1
a 1
S 2
1
1 r
1
2
  



15. Find the sum, if possible:
2 1 1 1
...
3 3 6 12
   
1 1
1
3 6
r
2 1 2
3 3
   1 r 1 Yes
    
1
2
a 4
3
S
1
1 r 3
1
2
  



16. Find the sum, if possible:
2 4 8
...
7 7 7
  
4 8
7 7
r 2
2 4
7 7
   1 r 1 No
    
NO SUM

17. Find the sum, if possible:
5
10 5 ...
2
  
5
5 1
2
r
10 5 2
   1 r 1 Yes
    
1
a 10
S 20
1
1 r
1
2
  



18. Thank You for Listening
