Pre-Algebra 12-2 Geometric Sequences Check 12-1 HOMEWORK
Dec 26, 2015
Pre-Algebra
12-2 Geometric Sequences
Students will be able to solve sequences and represent functions by completing the following assignments.
• Learn to find terms in an arithmetic sequence.• Learn to find terms in a geometric sequence.• Learn to find patterns in sequences.• Learn to represent functions with tables, graphs, or equations.
Pre-Algebra
12-2 Geometric Sequences
Today’s Learning Goal Assignment
Learn to find terms in a geometric sequence.
Pre-Algebra
12-2 Geometric Sequences12-2 Geometric Sequences
Pre-Algebra
Warm UpWarm Up
Problem of the DayProblem of the Day
Lesson PresentationLesson Presentation
Pre-Algebra
12-2 Geometric Sequences
Warm Up1. 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
Pre-Algebra
12-2 Geometric Sequences
Pre-Algebra
12-2 Geometric Sequences
Problem of the Day
Two students begin counting by 3’s at the same time. One counts up from 0, and the other counts down from 120. If each says one number every second, will both students ever say the same number at the same time?yes (60)
Pre-Algebra
12-2 Geometric Sequences
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.
Pre-Algebra
12-2 Geometric Sequences
Determine if the sequence could be geometric. If so, give the common ratio.
A. 1, 5, 25, 125, 625, …
Additional Example 1A: Identifying Geometric Sequences
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
Pre-Algebra
12-2 Geometric Sequences
Determine if the sequence could be geometric. If so, give the common ratio.
A. 2, 10, 50, 250, 1250, . . .
Try This: Example 1A
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
Pre-Algebra
12-2 Geometric Sequences
Determine if the sequence could be geometric. If so, give the common ratio.
B. 1, 3, 9, 12, 15, …
Additional Example 1B: Identifying Geometric Sequences
The sequence is not geometric.
Divide each term by the term before it.
1 3 9 12 15, . . .
54
43
33
Pre-Algebra
12-2 Geometric Sequences
Determine if the sequence could be geometric. If so, give the common ratio.
B. 1, 1, 1, 1, 1, . . .
Try This: Example 1B
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
Pre-Algebra
12-2 Geometric Sequences
Determine if the sequence could be geometric. If so, give the common ratio.
C. 81, 27, 9, 3, 1, . . .
Additional Example 1C: Identifying Geometric Sequences
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
Pre-Algebra
12-2 Geometric Sequences
Determine if the sequence could be geometric. If so, give the common ratio.
C. 2, 4, 12, 24, 96, . . .
Try This: Example 1C
The sequence is not geometric.
Divide each term by the term before it.
2 4 12 24 96, . . .
4232
Pre-Algebra
12-2 Geometric Sequences
Determine if the sequence could be geometric. If so, give the common ratio.
D. –3, 6, –12, 24, –48
Additional Example 1D: Identifying Geometric Sequences
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
Pre-Algebra
12-2 Geometric Sequences
Determine if the sequence could be geometric. If so, give the common ratio.
D. 1, 2, 4, 8, 16, . . .
Try This: Example 1D
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.
Pre-Algebra
12-2 Geometric Sequences
FINDING THE nth TERM OF A GEOMETRIC SEQUENCE
The nth term an of a geometric sequence with common ratio r is
an = a1rn–1.
Pre-Algebra
12-2 Geometric Sequences
Find the given term in the geometric sequence.
A. 11th term: –2, 4, –8, 16, . . .
Additional Example 2A: Finding a Given Term of a Geometric Sequence
an = a1rn–1
a11 = –2(–2)10 = –2(1024) = –2048
r = = –2 4–2
Pre-Algebra
12-2 Geometric Sequences
Find the given term in the geometric sequence.
A. 12th term: -2, 4, -8, 16, . . .
Try This: Example 2A
an = a1rn–1
a12 = –2(–2)11 = –2(–2048) = 4096
r = = –2 4–2
Pre-Algebra
12-2 Geometric Sequences
Find the given term in the geometric sequence.
B. 9th term: 100, 70, 49, 34.3, . . .
Additional Example 2B: Finding a Given Term of a Geometric Sequence
an = a1rn–1
a9 = 100(0.7)8 = 100(0.05764801) = 5.764801
r = = 0.7 70 100
Pre-Algebra
12-2 Geometric Sequences
Find the given term in the geometric sequence.
B. 11th term: 100, 70, 49, 34.3, . . .
Try This: Example 2B
a11 = 100(0.7)10 = 100(0.0282475249) 2.825
an = a1rn–1
r = = 0.7 70100
Pre-Algebra
12-2 Geometric Sequences
Find the given term in the geometric sequence.
C. 10th term: 0.01, 0.1, 1, 10, . . .
Additional Example 2C: Finding a Given Term of a Geometric Sequence
an = a1rn–1
a10 = 0.01(10)9 = 0.01(1,000,000,000) = 10,000,000
r = = 10 0.10.01
Pre-Algebra
12-2 Geometric Sequences
Find the given term in the geometric sequence.
C. 5th term: 0.01, 0.1, 1, 10, . . .
Try This: Example 2C
a5 = 0.01(10)4 = 0.01(10,000) = 100
an = a1rn–1
r = = 10 0.10.01
Pre-Algebra
12-2 Geometric Sequences
Find the given term in the geometric sequence.
D. 7th term: 1000, 200, 40, 8, . . .
Additional Example 2D: Finding a Given Term of a Geometric Sequence
an = a1rn–1
r = = 2001000
15
a7 = 1000( )6 = 1000( )= , or 0.064
15
8 125
1 15,625
Pre-Algebra
12-2 Geometric Sequences
Find the given term in the geometric sequence.
D. 12th term: 1000, 200, 40, 8, …
Try This: Example 2D
an = a1rn–1
r = = 2001000
15
a5 = 1000 ( )4 = 1000( )= , or 1.6
15
85
1 625
Pre-Algebra
12-2 Geometric Sequences
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?
Additional Example 3: Money Application
If Tara chooses $50 per sale, she will get a total of 10($50) = $500.
Pre-Algebra
12-2 Geometric Sequences
Additional Example 3 Continued
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.
Pre-Algebra
12-2 Geometric Sequences
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?
Try This: Example 3
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.
Pre-Algebra
12-2 Geometric Sequences
Lesson QuizDetermine 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