Math 6/7 NOTES 9.1
Name ____________________________________
Sequences
A sequence is an ordered list of numbers, with terms that follow a rule.

Each number in a sequence is called a ________________________.

The rule of the sequence describes the _______________________.

In an Arithmetic Sequence, the difference between any two consecutive terms is the
same. So, you can find the next term in the sequence by adding the same number (positive or
negative) to the previous term.
8 , 11 , 14 , 17 , 20 , 23
+3
+3
+3
+3
+3
29 , 23 , 17 , 11 , 5 , -1
+(-6) +(-6) +(-6) +(-6) +(-6)
o The difference between any two consecutive terms in arithmetic sequence is called the
____________________________________.
Continue the patterns below with the next three terms. What rule does each pattern follow?

1) 5, 10, 15, _____ , _____ , _____
Rule:____________________________________________________
2) 2, 4, 6, 8, _____ , _____ , _____
Rule:____________________________________________________
3) -100, -75, -50, _____ , _____ , _____
Rule:____________________________________________________
In a Geometric Sequence, the quotient of any two consecutive terms is the same. So, you
can find the next term in the sequence by multiplying the previous term by the same number
(whole or fraction) to the previous term.
3 , 6 , 12 , 24 , 48
·2
·2
·2
·2
4 , 2 , 1 ,
·
·
·
,
·
,
·
o The factor between any two consecutive terms in geometric sequence is called the
____________________________________.
Continue the patterns below with the next three terms. What rule does each pattern follow?

4) 2 , 4 , 8 , 16 , _____ , _____ , _____
Rule:____________________________________________________
5)
, , 1 , 2 , _____ , _____ , _____
Rule:____________________________________________________
6) -1 , 4 , -16 , _____ , _____ , _____
Rule:____________________________________________________
A sequence that does not have a common difference or common ratio is NEITHER an
arithmetic or geometric sequence.
1, 2, 4, 7, 11
+1 +2 +3 +4
Math 6/7 PRACTICE 9.1
Name ____________________________________
Examples: Identify the pattern as an arithmetic sequence, a geometric sequence, or neither. Then
describe the relationship (rule) between two consecutive terms.
1.)
2, 3, 4, 5, 6, …
________________________________
________________________________
2)
1, 2, 4, 8, 16, …
_________________________________
_________________________________
3.)
8, 13, 18, 23, 28, …
________________________________
________________________________
4)
5, 6, 8, 11, 15, …
_________________________________
_________________________________
5)
35, 28, 21, 14, …
________________________________
________________________________
6)
1, 3, 9, 27, …
_________________________________
_________________________________
7)
2, -4, 8, -16, …
________________________________
________________________________
8)
121, 1221, 12221, …
_________________________________
_________________________________
9)
-8, -6, -4, -2
________________________________
________________________________
10)
2, 6, 24, 120
_________________________________
_________________________________
11) What is the common ratio of the geometric sequence shown?:
8, 32, 128, …
12) What is the 7th term of the arithmetic sequence?:
-29, -21, -13, -5, …
13) Find the missing terms in the sequence:
26, 33, 40, ____, 54, 61, ____, …
Write the next three terms of the following sequences.
14) 0, 13, 26, 39, _____, _____, _____
15) 8, 5, 2, -1, -4, _____, _____, _____
16) -2, -6, -18, -54, _____, _____, _____
17) Create three patterns. Then describe the relationship (rule) between each consecutive term.
Arithmetic: ____________________________________________________ Rule: _________________________________
Geometric: ____________________________________________________ Rule: __________________________________
Neither:
____________________________________________________ Pattern: _______________________________
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5-10 Study Guide and Intervention
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Example
State whether the sequence is arithmetic, geometric, or neither. If it
is arithmetic or geometric, state the common difference or common ratio and
write the next three terms of the sequence.
a. 7, 14, 28, 56, 112, . . .
The common ratio is 2, so the sequence is geometric. The next three terms are
112 ? 2 or 224, 224 ? 2 or 448, and 448 ? 2 or 896.
b. 8, 6, 4, 2, 0, 22, . . .
The common difference is –2, so the sequence is arithmetic. The next three terms are
22 1 (22) or 24, 24 1 (22) or 26, and 26 1 (22) or 28.
Exercises
State whether each sequence is arithmetic, geometric, or neither. If it is arithmetic
or geometric, state the common difference or common ratio and write the next
three terms.
1. 10, 20, 40, 80, 160, . . .
25 25
2 4
2. 7, 17, 27, 37, 47, . . .
3. 100, 50, 25, }}, }}, . . .
4. 12, 9, 6, 3, 0, . . .
5. 1, 2, 2, 3, 3, 3, . . .
6. 1, –1, 1, –1, 1, . . .
7. 2, 3, 4.5, . . .
8. –21, –10, 1, 12, 23, . . .
9. 1.6, 3.2, 4.8, . . .
10. 4, 12, 36, . . .
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5-10 Skills Practice
State whether each sequence is arithmetic, geometric, or neither. If it is
arithmetic or geometric, state the common difference or common ratio and
write the next three terms of the sequence.
1. 5, 9, 13, 17, 21, . . .
2. 15, 10, 5, 0, –5, . . .
3. 1, 3, 9, 27, 81, . . .
4. 1, 0.3, 0.09, 0.027, 0.0081, . . .
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5. }}, 1, }}, 1, }}, . . .
6. 0.4, 1.6, 6.4, 25.6, 102.4, . . .
1 1 1
2 4 8
7. 2, 1, }}, }}, }}, . . .
8. 30, 21, 12, 3, –6, . . .
1 1 1 1
16 8 4 2
9. }}, }}, }}, }}, 1, . . .
10. 3, –4, –11, –18, –25, . . .
Lesson 5-10
11. 1, 2, 4, 7, 11, . . .
12. 0, 9, 18, 27, 36, . . .
13. 100, 10, 1, 0.1, 0.01, . . .
14. 1.0, 1.2, 1.4, 1.6, 1.8, . . .
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5-10 Reading to Learn Mathematics
How can sequences be used to make predictions?
Pre-Activity
Do the activity at the top of page 249 in your textbook. Write
your answers below.
a. What is the reaction distance for a car going 70 mph?
b. What is the braking distance for a car going 70 mph?
c. What is the difference in reaction distances for every 10-mph
increase in speed?
d. Describe the braking distance as speed increases.
Reading the Lesson
Write a definition and give an example of each new vocabulary word or phrase.
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1. sequence
2. arithmetic
sequence
3. term
4. common
difference
Lesson 5-10
5. geometric
sequence
6. common
ratio
7. What is the common difference in the sequence 8, 17, 26, 35, . . .?
8. What is the common ratio in the sequence 1, –2, 4, –8, 16, . . .?
Helping You Remember
9. Sequence is a word that is used in everyday English.
a. Find the definition of sequence in a dictionary. Write the definition.
b. Explain how the English definition can help you remember how sequence is used in
mathematics.
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5-10 Enrichment
Populations often grow according to a geometric sequence. If the population of a city grows
at the rate of 2% per year, then the common ratio, r, is 1.02. To find the population of a city
of 100,000 after 5 years of 2% growth, use the formula arn21, where r is the common ratio
and n is the number of years.
ar n21 5 100,000 (1.02)521
5 100,000
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< 108,243
So, the city has a population of 108,243 people.
After a few years, a small change in the annual growth rate can cause enormous differences
in the population.
Assume that the nation of Grogro had a population of one million in 2000. Using a
calculator, find the population of the country in the years 2020, 2070, and 2120 at
growth rates of 1%, 3%, and 5% per year. Record your results in the table below.
Population of Grogro
Year
Growth
Rate
1.
1%
2.
3%
3.
5%
2020
2070
2120
Suppose we want to find the total upward distance a bouncing ball has moved. It bounces
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up 36 inches on the first bounce and }} times its height on each of five more consecutive bounces.
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The distances form the terms of a geometric sequence. We want to find the sum of the
distances, or the sum of the six terms in the sequence.
The sum of the terms of a geometric sequence is called a geometric series. The formula for
a 2 ar n
the sum Sn of the first n terms of a geometric series is Sn 5 }}, where a 5 the first
12r
term and r 5 the common ratio (r Þ 1).
a 2 ar n
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Sn 5 }}
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So, the ball bounces upward approximately 118.4 inches.
Use the formula above to find each sum. Then check your answer by adding.
4. 5 1 10 1 20 1 40 1 80
5. 80 1 240 1 720 1 2160 1 6480
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