Upload
jody-curtis
View
231
Download
2
Embed Size (px)
Citation preview
Holt McDougal Algebra 2
Geometric Sequences and SeriesGeometric Sequences and Series
Holt Algebra 2Holt McDougal Algebra 2
• How do we find the terms of an geometric sequence, including geometric means?
• How do we find the sums of geometric series?
Holt McDougal Algebra 2
Geometric Sequences and Series
Serena Williams was the winner out of 128 players who began the 2003 Wimbledon Ladies’ Singles Championship. After each match, the winner continues to the next round and the loser is eliminated from the tournament. This means that after each round only half of the players remain.
Holt McDougal Algebra 2
Geometric Sequences and Series
The number of players remaining after each round can be modeled by a geometric sequence. In a geometric sequence, the ratio of successiveterms is a constant called the common ratio r (r ≠ 1) . For the players remaining, r is .
Holt McDougal Algebra 2
Geometric Sequences and Series
Recall that exponential functions have a commonratio. When you graph the ordered pairs (n, an) of ageometric sequence, the points lie on an exponentialcurve as shown. Thus, you can think of a geometricsequence as an exponential function with sequentialnatural numbers as the domain.
Holt McDougal Algebra 2
Geometric Sequences and Series
Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference.
Identifying Geometric Sequences
1. 100, 93, 86, 79, . . .
Diff. It could be arithmetic, with d = –7.
7 77
Ratio ,100
93,
93
86
86
79
2. 180, 90, 60, 15, . . .
Diff. It is neither.30 4590
Ratio ,180
90,
90
60
60
15
Holt McDougal Algebra 2
Geometric Sequences and Series
Determine whether the sequence could be geometric or arithmetic. If possible, find the common ratio or difference.
Identifying Geometric Sequences
3. 5, 1, 0.2, 0.04, . . .
Diff. It could be geometric, with r = 0.2.
8.0 16.04
Ratio ,5
1,
1
2.
2.
04.
4. 1.7, 1.3, 0.9, 0.5, . . .
Diff. 4. 4.4.
Ratio ,7.1
3.1,
3.1
9.0
9.0
5.0It could be arithmetic, with d = –0.4.
Holt McDougal Algebra 2
Geometric Sequences and Series
The nth term of an geometric sequence is
where a1 is the first term and r is the common ratio.
General Rule for Geometric Sequence
11
nn raa
To state a recursive rule of an geometric sequence, state the first term and the rule in the following form:
where a1 is the first term and r is the common ratio.
Recursive Rule for Geometric Sequence
11 ____; nn araa
Holt McDougal Algebra 2
Geometric Sequences and SeriesFinding the nth Term Given a Geometric Sequence
,243 ,081 36, 12, 4, .5
Write a rule for the nth term of the geometric sequence. Then find a
7.
4
12r 3
1 1 n
n raa
4
3
177 34 a 634
29167 a
7294
Holt McDougal Algebra 2
Geometric Sequences and SeriesFinding the nth Term Given a Geometric Sequence
,64
3 ,
32
3 ,
16
3 ,
8
3 ,
4
3 .6
Write a rule for the nth term of the geometric sequence. Then find a
7.
4/3
8/3r
2
1
1 1 n
n raa
4
3
2
1
17
7 2
1
4
3
a
256
3
Holt McDougal Algebra 2
Geometric Sequences and SeriesFinding the nth Term Given a Geometric Sequence
,10 ,1 ,1.0 ,01.0 ,001.0 .7
Write a rule for the nth term of the geometric sequence. Then find a
7.
001.0
01.0r 10
1 1 n
n raa
001.0
10
177 10001.0 a 1000
Holt McDougal Algebra 2
Geometric Sequences and SeriesFinding the nth Term Given Two terms of a Sequence
324 ,36 .8 53 aa
Write a rule for the nth term of the geometric sequence. Then find a
8.
3 5 35 raa
324 236r36 36
92 r3 r
1515 raa
324 41 3a
181324 a81 81
41 a
13 4 nna
188 34 a
8748 8748or
Holt McDougal Algebra 2
Geometric Sequences and SeriesFinding the nth Term Given Two terms of a Sequence
40 ,8 .9 54 aa
Write a rule for the nth term of the geometric sequence. Then find a
8.
4 5 45 raa
40 18r8 8
5r
1515 raa
40 41 5a
162540 a625 625
064.01 a
15064.0 nna 18
8 5064.0 a 5000
Holt McDougal Algebra 2
Geometric Sequences and SeriesWriting a Recursive Rule for an Geometric Sequence
,1 ,5 25, 125, 625, .10Write the recursive rule for the nth term of the sequence.
625
125r
5
1
6251 a
1 nn ara
5
1
Holt McDougal Algebra 2
Geometric Sequences and SeriesWriting a Recursive Rule for an Geometric Sequence
,432 ,81 27, ,9 3, .11 Write the recursive rule for the nth term of the sequence.
3
9r 3
31 a
1 nn ara
3
Holt McDougal Algebra 2
Geometric Sequences and Series
Lesson 5.3 Practice A