9-3 Geometric Sequences
Quick Review
In a geometric sequence, the ratio of consecutive terms is constant. This ratio is the common ratio.
For a geometric sequence, a is the first term,
a
n
eh sub n is the nth term, n is the number of the term, and r is the common ratio.
An explicit formula is
a
n
=
a
·
r
n
−
1
.
eh sub n , equals eh middle dot . r super n minus 1 end super . .
A recursive formula is
a
n
=
a
n
−
1
·
r
,
eh sub n , equals . eh sub n minus 1 end sub . middle dot r comma with
a
1
=
a
.
eh sub 1 , equals eh .
The geometric mean of two positive numbers x and y is
x
y
.
square root of x y end root , .
Example
What is the sixth term of the geometric sequence that begins 2, 6, 18, …?
a
1
=
2
and
r
=
6
÷
2
=
3
a
6
=
2
·
3
6
−
1
=
486
Substitute
6
for
n
,
2
for
a
1
,
and
3
for
r
.
table with 2 rows and 3 columns , row1 column 1 , eh sub 1 , column 2 equals 2 and r equals 6 divides 2 equals 3 , column 3 , row2 column 1 , eh sub 6 , column 2 equals 2 middle dot . 3 super 6 minus 1 end super . equals 486 , column 3 table with 2 rows and 1 column , row1 column 1 , cap substitute . 6 , for , n comma 2 , for , eh sub 1 , comma , row2 column 1 , and , 3 , for , r . , end table , end table
The sixth term is 486.
Exercises
Determine whether each sequence is geometric. If so, identify the common ratio and find the next two terms.
-
1
,
1
2
,
1
4
,
1
8
,
…
1 comma , 1 half . comma , 1 fourth , comma , 1 eighth , comma dot dot dot
- 1, 3, 5, 7, …
- 3, 3.6, 4.32, 5.184, ….
Find the missing term(s) of each geometric sequence.
-
3
,
□
,
12
,
…
3 comma white square comma 12 comma dot dot dot
-
0
.
004
,
□
,
0
.
4
,
…
0 . 004 comma white square comma 0 . 4 comma dot dot dot
-
−
20
,
□
,
□
,
□
,
−
1
.
25
,
…
negative 20 comma white square comma white square comma white square comma negative 1 . 25 comma dot dot dot
Write an explicit formula for each geometric sequence.
- 1, 2, 4, 8, …
-
25
,
5
,
1
,
1
5
,
…
25 comma 5 comma 1 comma , 1 fifth , comma dot dot dot
Use an explicit formula to find the 10th term of each geometric sequence.
- 5, 10, 20, 40, …
-
−
3
,
6
,
−
12
,
24
,
…
negative . 3 comma 6 comma negative 12 comma . 24 comma dot dot dot
9-4 Arithmetic Series
Quick Review
A series is the expression for the sum of the terms of a sequence.
An arithmetic series is the sum of the terms of an arithmetic sequence. The sum
S
n
s sub n of the first n terms of an arithmetic series is
s
n
=
n
2
(
a
1
+
a
n
)
.
s sub n , equals , n over 2 . open . eh sub 1 , plus , eh sub n . close . . You can use a summation symbol,
∑
,
sum comma and lower and upper limits to write a series. The lower limit is the least value of n and the upper limit is the greatest value of n.
Example
What is the sum of the arithmetic series?
2
+
5
+
8
+
11
+
14
+
17
+
20
2 plus 5 plus 8 . plus . 11 plus 14 plus 17 plus 20
a
1
=
2
,
a
7
=
20
,
and
n
=
7.
S
7
=
7
2
(
2
+
20
)
Substitute
7
for
n
,
2
for
a
1
,
and
20
for
a
7
.
=
77
Evaluate
.
table with 3 rows and 3 columns , row1 column 1 , eh sub 1 , column 2 equals 2 comma , eh sub 7 , equals 20 comma , column 3 and , n equals 7. , row2 column 1 , s sub 7 , column 2 equals , 7 halves . open , 2 plus 20 , close , column 3 cap substitute . 7 , for , n comma 2 , for , eh sub 1 , comma , and , 20 , for , eh sub 7 , . , row3 column 1 , , column 2 equals 77 , column 3 cap evaluate , . , end table
Exercises
Use summation notation to write each arithmetic series for the specified number of terms. Then evaluate the sum.
-
10
+
7
+
4
+
…
;
n
=
5
10 plus 7 plus 4 plus . dot dot dot semicolon , n equals 5
-
50
+
55
+
60
+
…
;
n
=
7
50 plus 55 plus 60 plus . dot dot dot semicolon , n equals 7
-
6
+
7
.
4
+
8
.
8
+
…
;
n
=
11
6 plus 7 . 4 plus 8 . 8 plus dot dot dot semicolon , n equals 11
-
21
+
19
+
17
+
…
;
n
=
8
21 plus 19 plus 17 plus . dot dot dot semicolon , n equals 8
Find the number of terms in each series, the first term, and the last term. Then evaluate the sum.
-
∑
n
=
1
3
(
17
n
−
25
)
sum , from , n equals 1 , to , 3 , of . open . 17 n minus 25 . close
-
∑
n
=
2
10
(
1
2
n
+
3
)
sum , from , n equals 2 , to , 10 , of . open . 1 half , n plus 3 . close
