A
geometric sequence
{
a
n
}
is a sequence that can be written in the following form.
•
a
1
=
a
•
a
n
=
ra
n

1
where
r
6
= 0. The number
r
is called the
common ratio
.
Example.
{
1
,
3
,
9
,
27
,
···}
is a geometric sequence with common ratio
r
=3.
Example.
{
1
,

1
,
1
,

1
,
···}
is a geometric sequence with common ratio
r
=

1.
Example.
Is the sequence
{
a
n
}
=
{
2
n
}
a geometric sequence?
Solution.
a
n
+1
a
n
=
2
n
+1
2
n
= 2. Since
a
n
+1
a
n
is a constant, the sequence is a geometric sequence with
r
=2.
Example.
Is the sequence
{
n
+1
n
}
=
{
a
n
}
a geometric sequence?
Solution.
a
n
+1
a
n
=
n
+2
n
+1
n
+1
n
=
n
(
n
+2)
(
n
+1)
2
. Since
a
n
+1
a
n
is not constant, the sequence is not a geometric
sequence.
Finding the nth term of a geometric sequence:
a
1
=
a
a
2
=
ar
a
3
=
ar
2
···
a
n
=
ar
n

1
The above shows that the
n
th term,
a
n
, of a geometric sequence is
•
a
n
=
ar
n

1
Sum of a geometric sequence:
Let
S
n
be the sum of the ﬁrst
n
terms of a geometric sequence. Then
S
1
=
a
1
S
2
=
a
+
ar
=
a
(1 +
r
)
S
3
=
a
+
ar
+
ar
2
=
a
(1 +
r
+
r
2
)
···
S
n
=
a
+
ar
+
ar
2
+
···
+
ar
n

1
=
a
(1 +
r
+
r
2
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '11
 Kutter
 Algebra, Euclid's Elements, $10, Geometric progression, $100

Click to edit the document details