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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
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 Fall '11
 Kutter
 Algebra

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