Download Geometric Sequences

8.3 – Geometric Sequences
 A sequence is geometric if consecutive numbers always
have the same common ratio (r).
 Ex: 2, 4, 8, 16, 32, …
 Ex:
2 4 8
16
,  , ,  , ...
3 9 27 81
has a common ratio of 2
4 8 16 32
 

2 4 8 16
has a common ratio of -2/3
 Geometric sequences are always of the form an = a1 rn-1
 r = the common ratio
 The sequence will be a1, a1r, a1r2, a1r3, …, a1rn-1
 Ex: Find a formula for the following geometric sequence,
then find a9: 5, 15, 45, …
 an = a1rn-1 = 5(3)n-1
 a9 = 5(3)8 = 32805
 Ex: If the 4th term of a geometric sequence is 125 and the
10th term is 125/64, find the 14th term.
 Think about the relationship between the 10th and 4th
terms!
a10  a4 r
6
125
a14 
1024
125
 125r 6
64
1
 r6
64
125  1 
a14 
 
64  2 
4
1
r
2
a14  a10 r 4
 The sum of a finite geometric sequence is:
 a1
 1 rn 
Sn  a1 

1 r 

st
= 1 term being summed
 n = # terms being summed
 r = common ratio
8
 Ex: Find the sum:
k
3(0.6)

k 1
 n = 8, a1 = 3(0.6) = 1.8, r = 0.6
 1  0.68 
Sn  1.8 
  1.8(2.458)  4.424
 1  0.6 
 The sum of an infinite geometric sequence is:
a1
S
1 r
 If |r| ≥ 1 , the series does not have a sum.

 Ex: Find the sum:
k 1
4(0.6)

k 1
 Use the infinite sum formula!
 To find a1, evaluate for k = 1

a1 = 4(0.6)1-1 = 4(0.6)0 = 4
4
4
 10
Sn 

1  0.6
0.4
Find the sum of the series.
1
16  

2
n1
10
0%
0%
0%
0%
0%
19
.9
8
5.
.0
3
4.
15
.9
8
3.
31
.9
7
2.
16
31.97
15.98
.03
19.98
16
1.
n
0%
11
74
5.
0%
0%
0%
0%
70
96
4.
21
41
3.
66
70
2.
1174
1249
6670
2141
7096
12
49
1.
The population in Dodge City
decreases by 6% yearly. If the
population was 1600 in 1875, what
was the population in 1880?
Find the sum of the series.
4
2 

n 0  5 
12
0%
0%
0%
0%
0%
9.
31
5.
9.
45
4.
7.
45
3.
7.
56
2.
10
7.56
7.45
9.45
9.31
10
1.
n
Find the sum of the infinite series.
9
4
, 3, 2, , ...
2
3
0%
0%
1/
2
0%
13
0%
9
0%
1/
2
5.
1
4.
-9
3.
5/
6
2.
65/6
-9
3/2
9
27/2
10
1.