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1. Determine if the sequence could be arithmetic. If so, give the common difference. 100, 50, 25, 12.5, . . .
Find the given term in each arithmetic sequence.
2. 12th term: a1 = 30, d = 0.5
3. 55th term: 4, 28, 52, 76
no
35.5
1300
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Pre-Algebra
12-212-2
Geometric Sequences
LearnLearn to find terms in a geometric to find terms in a geometric sequencesequence..
geometric sequencegeometric sequence
common ratiocommon ratio
VocabularyVocabulary
In a geometric sequence, the ratio of one term to the next is always the same. This ratio is called the common ratio. The common ratio is multiplied by each term to get the next term.
Determine if the sequence could be geometric. If so, give the common ratio.
A. 1, 5, 25, 125, 625, …
The sequence could be a geometric with a common ratio of 5.
Divide each term by the term before it.
1 5 25 125 625, . . .
5555
Example: Identifying Geometric Example: Identifying Geometric SequencesSequences
Determine if the sequence could be geometric. If so, give the common ratio.
B. 1, 3, 9, 12, 15, …
The sequence is not geometric.
Divide each term by the term before it.
1 3 9 12 15, . . .
54
43
33
Example: Identifying Geometric Example: Identifying Geometric SequencesSequences
Determine if the sequence could be geometric. If so, give the common ratio.
C. 81, 27, 9, 3, 1, . . .
The sequence could be geometric with a common ratio of .1
3
Divide each term by the term before it.
81 27 9 3 1, . . .
13
13
13
13
Example: Identifying Geometric Example: Identifying Geometric SequencesSequences
Determine if the sequence could be geometric. If so, give the common ratio.
D. –3, 6, –12, 24, –48
The sequence could be geometric with a common ratio of –2.
Divide each term by the term before it.
–3 6 –12 24 –48, . . .
–2–2–2–2
Example: Identifying Geometric Example: Identifying Geometric SequencesSequences
Determine if the sequence could be geometric. If so, give the common ratio.
A. 2, 10, 50, 250, 1250, . . .
The sequence could be a geometric with a common ratio of 5.
Divide each term by the term before it.
2 10 50 250 1250, . . .
5555
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Determine if the sequence could be geometric. If so, give the common ratio.
B. 1, 1, 1, 1, 1, . . .
The sequence could be a geometric with a common ratio of 1.
Divide each term by the term before it.
1 1 1 1 1, . . .
1111
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Determine if the sequence could be geometric. If so, give the common ratio.
C. 2, 4, 12, 24, 96, . . .
The sequence is not geometric.
Divide each term by the term before it.
2 4 12 24 96, . . .
4232
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Determine if the sequence could be geometric. If so, give the common ratio.
D. 1, 2, 4, 8, 16, . . .
The sequence could be geometric with a common ratio of 2.
1 2 4 8 16, . . .
2222
Divide each term by the term before it.
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FINDING THE FINDING THE nnthth TERM OF A GEOMETRIC TERM OF A GEOMETRIC SEQUENCESEQUENCE
The The nnthth term an of a geometric sequence with term an of a geometric sequence with common ratio common ratio rr is is
aann = = aa11rrnn–1–1..
Find the given term in the geometric sequence.
A. 11th term: –2, 4, –8, 16, . . .
an = a1rn–1
a11 = –2(–2)10 = –2(1024) = –2048
r = = –2 4–2
Example: Finding a Given Term of Example: Finding a Given Term of a Geometric Sequencea Geometric Sequence
Find the given term in the geometric sequence.
B. 9th term: 100, 70, 49, 34.3, . . .
an = a1rn–1
a9 = 100(0.7)8 = 100(0.05764801) = 5.764801
r = = 0.7 70 100
Example: Finding a Given Term of Example: Finding a Given Term of a Geometric Sequencea Geometric Sequence
Find the given term in the geometric sequence.
C. 10th term: 0.01, 0.1, 1, 10, . . .
an = a1rn–1
a10 = 0.01(10)9 = 0.01(1,000,000,000) = 10,000,000
r = = 10 0.10.01
Example: Finding a Given Term of Example: Finding a Given Term of a Geometric Sequencea Geometric Sequence
Find the given term in the geometric sequence.
D. 7th term: 1000, 200, 40, 8, . . .
an = a1rn–1
r = = 2001000
15
a7 = 1000( )6 = 1000( )= , or 0.064
15
8 125
1 15,625
Example: Finding a Given Term of Example: Finding a Given Term of a Geometric Sequencea Geometric Sequence
Find the given term in the geometric sequence.
A. 12th term: -2, 4, -8, 16, . . .
an = a1rn–1
a12 = –2(–2)11 = –2(–2048) = 4096
r = = –2 4–2
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Find the given term in the geometric sequence.
B. 11th term: 100, 70, 49, 34.3, . . .
a11 = 100(0.7)10 = 100(0.0282475249) 2.825
an = a1rn–1
r = = 0.7 70100
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Find the given term in the geometric sequence.
C. 5th term: 0.01, 0.1, 1, 10, . . .
a5 = 0.01(10)4 = 0.01(10,000) = 100
an = a1rn–1
r = = 10 0.10.01
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Find the given term in the geometric sequence.
D. 12th term: 1000, 200, 40, 8, …
an = a1rn–1
r = = 2001000
15
a5 = 1000 ( )4 = 1000( )= , or 1.6
15
85
1 625
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Tara sells computers. She has the option of earning (1) $50 per sale or (2) $1 for the first sale, $2 for the second sale, $4 for the third sale and so on, where each sale is worth twice as much as the previous sale. If Tara estimates that she can sell 10 computers a week, which option should she choose?
If Tara chooses $50 per sale, she will get a total of 10($50) = $500.
Example: Money ApplicationExample: Money Application
a10 = ($1)(2)9 = ($1)(512) = $512
Option 1 gives Tara more money in the beginning, but option 2 gives her a larger total amount.
If Tara chooses the second option, her earnings for just the 10th sale will be more that the total of all the earnings in option 1.
Example ContinuedExample Continued
A gumball machine at the mall has 932 gumballs. If 19 gumballs are bought each day, how many gumballs will be left in the machine on the 7th day?
a7 = (932)(0.98)6 (932)(0.89) 829
an = a1rn–1
r = 0.98913932
n = 7 a1 = 932
There will be about 829 gumballs in the machine after 7 days.
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Determine if each sequence could be geometric. If so, give the common ratio.
1. 200, 100, 50, 25, 12.5, . . .
2. 4, 8, 12, 16, . . .
Find the given term in each geometric
sequence.
3. 7th term: , 1, 3, 9, . . .
4. 20th term: a1 = 800, r = 0.8
no
243
≈ 11.53
13
yes; 12
Lesson QuizLesson Quiz
