9-3: Geometric Sequences
Mr. Gallo
CP Algebra 2
Geometric Sequences
Given the sequence 5, 10, 20, 40, 80, …
 Is there a constant ratio of terms?
x2
 What are the next three numbers in the sequence?
160 _______,
320 ________
640
______,
Given the sequence 48, 72, 108, 162, 243, ….
 What is the constant ratio of terms? x1.5
 What are the next three numbers in the sequence?
364.5 ________,
546.75 ________
820.125
________,
A Geometric Sequence is formed by
previous term by a constant.
multiplying the ____________
1
Recursive formula for a Geometric
Sequence
Recursive formula for a Geometric Sequence:
The sequence formed by the recursive formula
 g1

 g n  rg n 1 For Integers n  2,
where r is a nonzero constant, is the geometric, or
exponential, sequence with first term and constant
multiplier.
When
r  1 the terms
increase
When
r 1
decrease
the terms
Example 1
Give the first six terms of the geometric
sequence defined by the recursive formula:
 g1  3

 g n  5 g n 1
Given: g n  rg n 1
for integers n  2
g
Then: g n  r
n 1
Term
1
2
Constant
5
Ratio = _______
3
15
3
4
5
6
75 375 1875 9375
2
Explicit Formula for a Geometric Sequence
Explicit Formula for a Geometric
Sequence:
In the geometric sequence with first term g1
and constant ratio r ,
g n  g1  r 
n 1
for integers
g
n >1
Example 2: Write the first five terms of the
sequence defined by: g  4  3 n 1
n
g1  4  3 
11
g 2  4  3
2 1
4
g 3  4  3
 12
g 4  4  3
31
4 1
 36
g 5  4  3
5 1
 324
 108
Example 3
 Write the following formulas for the
sequence 8, 4, 2, 1, ½, ¼, …
a) Recursive formula
 g1  8


1
 g n  2 g n 1 for n  2
b) Explicit formula
1
gn  8  
2
n 1
3
Suppose a ball is dropped from a height of 100 ft. and
bounces to 80% of its previous height after each bounce.
Find the height the ball reaches after the tenth bounce. ( A
bounce is counted when the ball hits the ground.) Let hn
be the greatest height of the ball after the nth bounce.
80%  .8 Initial height (h0) = ___________
100
Constant ratio __________
 
100 .8  80
h1 = _________________
Geometric Sequence:
Height
n 1
80 .8 
hn  h1  r = ____________________
n1
Constant ratio
h1
.8
h2
.8
h10
.8
hn
.8
Result (hn)
80
64
10.73
.8  hn 1 
Geometric Mean
What are the possible values of the missing term of
the geometric sequence: …, 28, ?, 7, …
In a geometric sequence, the middle term of any
three consecutive terms is equal to the square
root of the product of the other two. This is called
the geometric mean.
geometric mean  28  7  196  14
The possible values are the geometric mean and
its opposite. So the possible values are -14 and 14.
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Homework: p.584 #7-25 odd, 33-37 odd, 49
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