11.3
Geometric Sequences
1
Geometric Sequences
A geometric sequence is one in which each
term after the first is found by multiplying the
previous term by a constant called the common
ratio, r.
5, 10, 20, 40, 80, …
5 ● 2 = 10
10 ● 2 = 20
20 ● 2 = 40
2
Find the next 2 terms of the given geometric
sequence.
1
1
16, 8, 4, 2, 1, …,
2 4
3
The common ratio (r) can be found by dividing a
term by its previous term:
Given the geometric sequence:
a1 , a2 , a3 , a4 ,..., an
a3
a2
 r,
 r,
a1
a2
an
r
an 1
4
Formula for the nth term (explicit formula) of a
Geometric sequence:
an  a1  r
( n 1)
Write the formula for the nth term of the geometric
sequence and find the 15th term.
2, -6, 18, -54, …
a1  2
a15  2  (3)
n  15
a15  9,565,938
6 18 54
r


 3
2 6 18
14
5
The 9th term of a geometric sequence is 135.
If the common ratio is 3, find the 1st term.
( n 1)
an  135
n9
a1  ?
r 3
an  a1  r
(9 1)
135  a1  3
135  a1  6561
135
 a1
6561
5
a1 
243
6
In a geometric sequence, a4 = 6 and a8 = 96,
find a1 and the nth term.
an  a1  r
6  a1  r
6

a
1
3
r
(4 1)
( n 1)
96  a1  r
96
 a1
7
r
6 96

3
7
r
r
7
3
6r  96r
4
r  16
r2
(8 1)
(4 1)
6  a1  2
6  a1  8
3
 a1
4
3 ( n 1)
an   2
4
7
Geometric Mean
The square root of the product of 2 numbers or
the missing term(s) between 2 nonconsecutive
terms in a geometric sequence.
Find the missing term of the geometric sequence:
6 2 12
6, ____,
x 12

6 x
x 2  6  12
x  6  12
x6 2
8
The geometric mean of any two positive
numbers is the positive square root of the
product of the two numbers.
Find the geometric mean of
40
20 and 80
3 and 18.75
7.5
a1 and a3
a1 a3
Find the missing term of each geometric sequence:
25, ___,
50 100
36 ___,
9, ___,
18 ___,
72 144
9
Find the missing terms of the geometric sequence:
8 ____,
16 32
25, ____,
10 ____,
4 ____,
5
25
.
125
32
 25  r 5
125
32
 r5
125 25
2
r
5
10