Holt McDougal Algebra 2 Geometric Sequences and Series Holt Algebra 2Holt McDougal Algebra 2 How do we find the terms of an geometric sequence, including

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?


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.



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 .



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.



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



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.



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


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



,64

3 ,

32

3 ,

16

3 ,

8

3 ,

4

3 .6


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



,10 ,1 ,1.0 ,01.0 ,001.0 .7


7.

001.0

01.0r 10

1 1 n

n raa

001.0

10

177 10001.0 a 1000


Geometric Sequences and SeriesFinding the nth Term Given Two terms of a Sequence

324 ,36 .8 53 aa


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


Geometric Sequences and SeriesFinding the nth Term Given Two terms of a Sequence

40 ,8 .9 54 aa


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


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


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



Lesson 5.3 Practice A

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Holt McDougal Algebra 2 Geometric Sequences and Series Holt Algebra 2Holt McDougal Algebra 2 How do we find the terms of an geometric sequence, including