unit 8 mid checkpiont apply

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Cassidy Williams
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% Unit 8: Mid Checkpoint Apply
Unit 8: Inferences and Conclusions from Data Mid Checkpoint Apply
Interactive Tools
Unit 8: Inferences and Conclusions from Data Mid Checkpoint Apply
Lesson 1 – 2
Algebra 2
Please complete the following questions. It is important that you show all work you did to solve the
problems when you submit your work. This includes any calculations, diagrams, or graphs that
helped you solve it.
1. OPEN ENDED Write an arithmetic series for which 55= 10.
2. OPEN ENDED Write a geometric series for which r= 1 and n = 4.
3. Writing in Math Use the information on page 649 to explain how arithmetic series apply to
amphitheaters. Explain what the sequence and the series that can be formed from the given
numbers represent, and show two ways to find the seating capacity of the amphitheater if it has ten rows of seats.
4. Writing in Math Use the information on page 663 to explain how e-mailing a joke is related to a geometric series. Include an
explanation of how the situation could be changed to make it better to use a formula than to add terms.
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69. ACT/SAT What is the first term in the
arithmetic sequence?
American
High School
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A3
70. REVIEW The figures below show a
pattern of filled leshiti
circles that canesch
648 – 649 / 1100
relationship between 2 variables.
OYO
0
Ooo
OO
a a
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Which rule relates to the number of
wluite circles, to f, the number of dark
cireles?
F = 3/ H = 2/+1
GF=32-1
Main Ideas
. Find sums of
arithmetic series
. Use signa notion
100%
a
C 10
Minimize
D 11
New Vocabulary
series
arithmetic series
sigma notation
Index of summuti
GET READY for the Lesson
Austin, Texas has a strong musical
tradition. It is home lo many indoor
outdoor music venues where new and
established musicians perform regularly
Some of these venues are amphitheaters
that generally get wider as the distance
from
the stage increases
Suppose a section of an amphitheater can
seat 18 people in the first row and each
row can seat 4 more people than the
previous row,
Arithmetic Series The numbers of seats in the rows of the amphitheater
form an arithmetic sequence. To find the number of people who could sit
in the first four rows, add the first four terms of the sequence. That sum
is 18 + 22 +26 + 30 or 96. A series is an indicated sum of the terms of a
sequence. Since 18, 22, 26, 30 is an arithmetic sequence, 18 + 22 – 26 +30
is an arithmetic series.
S represents the sum of the first w terms of a series. For example, is the
sum of the first four terms.
Study Tip
*
Spiral Review
Find the exact solution(s) of each system of equations, lesson 11-7)
71. x + 2y = 33
72 ** + 2y = 33
x2 + y2 – 19 = 2x
x2 – y2 – 9
Write each equation in standard form. State whether the graph of the
equation is a parabala, circle, ellipse, or hyperbola. Then graph the equation.
Les 10
73. y – 3x + 6y + 12 = 0
74. *? – 143 + 4 = 9y? -36y
75. If y varies directly as x and y = 5 when x = 2, find y when x = 6. fleson 4
Simplify each expression – 0
77. +3.14
78.
130
53
42774
Find all the zeros of each function, flesson 6)
79. x) = -32 +222 +21
80. y(x) = 12x + 42-32-3
81. SAVINGS Mackenzie has 557 in her bank account. She begins receiving a
weekly allowance of $15, of which she deposits 20% in her bank account.
Write an equation that represents how much money is in Mackenzie’s
account after x weeks. Leson2-4
GET READY for the Next Lesson
PREREQUISITE SKILL Evaluate each expression for the given values of the variable.
Lesson –
82. 3n-1,-1,2,3,4
85. 6-0 =1,2,3,4
84.4+7;m – 1,2,3,4,5
85. 4 – 2kk-3,4,5,6,7
Indicated Sum
The sum of a series is
the result when the
terms of the series are
added. An indicated
Sun is the expression
that ilustrates the
Seris, which includes
the term + –
76.39
39
To develop a formula for the sum of any arithmetic series, consider the
series below. .
S-4+ 11 +18 + 25 +32 +39 +46 +53 +60
Write Sy in two different orders and add the two equations.
S = 4 + 11 + 18 + 25 + 32 +39 +46 +53 +60
(+) 5 = 60 +53 +46 +39 + 32 + 25 + 18 + 11 + 4
25-64 +64 +64 +64 +64 +64 +64 +64 +64
25-964)
Note that the sum had 9 terms.
5:01)
The sum of the first and last terms of the series is 61.
An arithmetic series S, has n terms, and the sum of the first and last terms
is + . Thus, the formula S. = 24 + .) represents the sum of any
arithmetic series
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662 – 663 / 1100
American
High School
e sequences, the later the term in
You may have noticed that in some describe this is the
the sequence, the closer the value is
as increases, approaches 0. The value that the terms of a sequence
approach, in this case, is called the limit of the sequence. Other types of
infinite sequences may also have limits. If the terms of a sequence do not
approach a unique value, we say that the limit of the sequence does not exist.
Main Ideas
Hind sunod
geometri
. Find spects of
geometric series
GET READY for the Lesson
100%
Suppose you e-mail a joke to three friends
O
friends sends the joke on to three of
person who receives the joke on Tuesday sends it to three more people
on Wednesday, and so on.
New Vocabulary
some series
E-Mail Jokes
DOX
< ACTIVITY 1 Find the limit of the geometric sequence 1, 39. Step 1 Enter the sequence. • The formula for this sequence is = • Position the cursor on Ll in the STATEDIT Edit screen and enter the formula seq.N.N.1,10,1). This generates the values 1, 2...., 10 of the index N. • Position the cursor on L2 and enter the formula seql(1/3)^{N-1).N,1,10,1). This generates the first ten terms of the sequence KEYSTROKES: Review segniences in the Grupining Calculator Lab on page 632. Notice that as increases, the terms of the given sequence get closer and closer too. If you scroll down, you can see that for 2 & the terms are so close to that the calculator expresses them in scientific notation. This suggests that the limit of the sequence is O. Step 2 Graph the sequence. • Use a STAT PLOT to graph the sequence. Use ll as the list and L2 as the Ylist. KEYSTROKES: Review STAT PLOTS OR page 92 The graph also shows that, as n increases, the terms approach 0. In fact, for 26, the marks appear to lie on the horizontal axis. This strongly suggests that the limit of the sequence is 0. Geometric Series Notice that every day, the number of people who read your joke is three times the number that read it the day before. By Sunday, the number of people, including yourself, who have read the joke is 1+3+ 9 + 27 +81 +243 + 729 + 2187, or 3280! The numbers 1,3,4, 27, 41, 243,729, and 2187 form a geometric sequence in whicha Tandr 3. The indicated sum of the numbers in the sequence, 1+3+9+27 +81 +243 + 729 +2187, is called a geometric series >>
To develop a formula for the sum of a geometrie series, consider the series
given in the e-mail situation above. Multiply cach term in the series by
the common ratio and subtract the result from the original series.
S8 = 1 + 3+9+27 +81 +243 +229 +2157
(-)35 = 3+9+27+81 +243 +229 +2187 +6561
(1-3) = 1 + 0 + 0 + 0 + 0 + 0 + 0+ 0-6561
first term in series
EXERCISES
Use a graphing calculator to find the limit, if it exists, of each sequence.
last term in series multiplied by
common ratio; in this case,
Study Tip
Sy=1-3 or 3280
common ratio
1… =)
2.1=(-)
221
Terms of
Geometric
Sequences
Remember that can
also be written aan
5. a, =
The expression for can be written as 5;=”ITA rational
expression like this can be used to find the sum of any geometric series
A
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