Geometric-Sequences-and-Seriesssssss.ppt

geometric sequence and series

1. 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?

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

3. A sequence is geometric if the ratios of
consecutive terms are the same.
Geometric Sequence
32 4
1 2 3
.....
aa a
r
a a a
   
The number r is the common ratio ratio.

4. 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 2
n
Yes
4(3)
n
No n
2
No
(-1)
n
/3

5. Finding the nth term of a
Geometric Sequence
a
n
= a
1
r
n – 1
r
a
a

2
1

6. Ex. 2b
Write the first five terms of the
geometric sequence whose first
term is a
1 = 9 and r = (1/3).
931
1
3
1
9
,,,,

7. Ex. 3Find the 15Find the 15
thth
term of the geometric term of the geometric
sequence whose first term is 20 and sequence whose first term is 20 and
whose common ratio is 1.05whose common ratio is 1.05
a
n = a
1r
n – 1
a
15 = (20)(1.05)
15 – 1
a
15 = 39.599

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

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

10. Ex. 6
Find the sum of the first 12 terms of the Find the sum of the first 12 terms of the
series 4(0.3series 4(0.3
))nn
= 4(0.3)
1
+ 4(0.3)
2
+ 4(0.3)
3
+ … + 4(0.3)
12
r
raa
S
n
n



1
11
3.01
)3.0(2.12.1
12
12


S = 1.714

11. Ex. 7
Find the sum of the first 5 terms of the Find the sum of the first 5 terms of the
series 5/3 + 5 + 15 + …series 5/3 + 5 + 15 + …
r = 5/(5/3) = 3
r
raa
S
n
n



1
11
31
)3(
3
5
3
5
5
5


S
= 605/3

12. 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



13. Find the sum, if possible:
1 1 1
1 ...
2 4 8
   
1 1
12 4
r
11 2
2
   1 r 1 Yes    
1a 1
S 2
11 r
1
2
  



14. Find the sum, if possible:
2 1 1 1
...
3 3 6 12
   
1 1
13 6
r
2 1 2
3 3
   1 r 1 Yes    
1
2
a 43
S
11 r 3
1
2
  



15. 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    
NOSUM

16. Find the sum, if possible:
5
10 5 ...
2
  
5
5 12
r
10 5 2
   1 r 1 Yes    
1a 10
S 20
11 r
1
2
  



17. The Bouncing Ball Problem – Version A
A ball is dropped from a height of 50 feet. It rebounds 4/5 of
it’s height, and continues this pattern until it stops. How far
does the ball travel?
50
40
32
32/5
40
32
32/5
40
S 45
50
4
1
0
1
55
4
 




18. The Bouncing Ball Problem – Version B
A ball is thrown 100 feet into the air. It rebounds 3/4 of
it’s height, and continues this pattern until it stops. How far
does the ball travel?
100
75
225/4
100
75
225/4
10
S 80
100
4 4
3
1
0
1
0
3
 




19. The sum of the first n terms of a sequence is
represented by summation notation.
Definition of Summation Notation
1 2 3 4
1
n
i n
i
a a a a a a

      
index of
summation
upper limit of summation
lower limit of summation
5
1
4
n
n

1 2 3 4 5
4 4 4 4 4   
4 16 64 256 1024    
1364

20. 
12
1
Example 6. Find the sum 4 0.3
n
n

Write out a few terms.
 
12
1 2 3 12
1
4 0.3 4 0.3 4 0.3 4 0.3 ... 4 0.3
n
n
    

1
4 0.3 0.3 and 12a r n   

12
1
1
1
4 0.3
1
n
n
n
r
a
r

 
 
 
 

12
1 0.3
4 0.3
1 0.3
 
  
  
1.714
If the index began at i = 0, you would have
to adjust your formula
 
12 12
0
0 1
4 0.3 4 0.3 4 0.3
n
n
i n 
   
12
1
4 4 0.3
n
n
  4 1.714 5.714  

21. 0
n
b
3
6
5


 

 
 

0
3
6
5
 
 
 
1
3
6
5
 

 
 
2
3
6
5
 

 
 
...
1a
S
1 r


6
15
3
1
5
 
