Problem of the Day

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Warm Up
Problem of the Day
Lesson Presentation
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Warm Up Use the table to write an equation. 1.
2. 3.
x 1 2 3 4
y 5 10 15 20
y 5x
x 1 2 3 4
y 2.5 5 7.5 10
y 2.5x
x 1 2 3 4
y 5 8 11 14
y 3x 2
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Problem of the Day A movie ticket at a certain
theater costs 4.50 for a child and 8.75 for an
adult. How much will it cost for a family of two
adults and three children to see a movie?
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Learn to identify and evaluate arithmetic
sequences.
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Vocabulary
sequence term arithmetic sequence common
difference
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Warm Up Find the next two numbers in the pattern,
using the simplest rule you can find. 1. 1, 5,
9, 13, . . . 2. 80, 87, 94, 101, . . . 3. 3, 9,
7, 13, 11, . . .
17, 21
108, 115
17, 15
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A sequence is an ordered list of numbers or
objects, called terms. In an arithmetic sequence,
the difference between one term and the next is
always the same. This difference is called the
common difference. The common difference is added
to each term to get the next term.
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Additional Example 1A Identifying Arithmetic
Sequences
Determine if the sequence could be arithmetic. If
so, give the common difference. 5, 8, 11, 14, 17,
. . .
The terms increase by 3.
5 8 11 14 17, . . .
3
3
3
3
The sequence could be arithmetic with a common
difference of 3.
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Additional Example 1B Identifying Arithmetic
Sequences
Determine if the sequence could be arithmetic. If
so, give the common difference. 1, 3, 6, 10, 15,
. . .
Find the difference of each term and the term
before it.
1 3 6 10 15, . . .
5
4
3
2
The sequence is not arithmetic.
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Additional Example 1C Identifying Arithmetic
Sequences
Determine if the sequence could be arithmetic. If
so, give the common difference. 65, 60, 55, 50,
45, . . .
The terms decrease by 5.
65 60 55 50 45, . . .
5
5
5
5
The sequence could be arithmetic with a common
difference of 5.
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Additional Example 1 Finding the Common
Difference in an Arithmetic Sequence
Find the common difference in each arithmetic
sequence.
A. 11, 9, 7, 5,
11, 9, 7, 5,
The terms decrease by 2.
2 2 2
The common difference is -2.
B. 2.5, 3.75, 5, 6.25,
2.5, 3.75, 5, 6.25,
The terms increase by 1.25.
1.25 1.25 1.25
The common difference is 1.25.
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Check It Out Example 1
Find the common difference in each arithmetic
sequence.
A. 2, 6, 10, 14,
2, 6, 10, 14,
The terms increase by 4.
4 4 4
The common difference is 4.
B. 2.5, 5, 7.5, 10,
2.5, 5, 7.5, 10,
The terms increase by 2.5.
2.5 2.5 2.5
The common difference is 2.5.
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Additional Example 2 Finding Missing Terms in an
Arithmetic Sequence
Find the next three terms in the arithmetic
sequence 8, 3, 2, 7, ...
Each term is 5 more than the previous term.
7 5 12
Use the common difference to find the next three
terms.
12 5 17
17 5 22
The next three terms are 12, 17, and 22.
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Check It Out Example 2
Find the next three terms in the arithmetic
sequence 9, 6, 3, 0, ...
Each term is 3 more than the previous term.
0 3 3
Use the common difference to find the next three
terms.
3 3 6
6 3 9
The next three terms are 3, 6, and 9.
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Additional Example 3A Identifying Functions in
Arithmetic Sequences
Find a function that describes each arithmetic
sequence.
6, 12, 18, 24,
n n 6 y
1
2
3
4
n
1 6
Multiply n by 6.
6
2 6
12
3 6
18
y 6n
4 6
24
n 6
6n

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Additional Example 3B Identifying Functions in
Arithmetic Sequences
Find a function that describes each arithmetic
sequence.
4, 8, 12, 16,
n n ( 4) y
1
2
3
4
n
1 (4)
4
Multiply n by -4.
2 (4)
8
y 4n
3 (4)
12
4 (4)
16
n (4)
4n

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Check It Out Example 3A
Find a function that describes each arithmetic
sequence.
3, 6, 9, 12,
n n 3 y
1
2
3
4
n
1 3
Multiply n by 3.
3
2 3
6
3 3
9
y 3n
4 3
12
n 3
3n

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Check It Out Example 3B
Find a function that describes each arithmetic
sequence.
7, 14, 21, 28,
n n (-7) y
1
2
3
4
n
1 (-7)
-7
Multiply n by -7.
2 (-7)
-14
y -7n
3 (-7)
-21
4 (-7)
-28
n (-7)
-7n

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Additional Example 4 Application
A DVD costs 3.95 to rent, plus 0.45 for each
day it is returned late. Find a function that
describes the arithmetic sequence, and then find
the total cost of renting the DVD and returning
it 9 days late.
n 3.95 0.45n y
1
2
3
4
n
Multiply n by 0.45, and then add the 3.95
rental fee.
3.95 0.45(1)
4.40
3.95 0.45(2)
4.85
3.95 0.45(3)
5.30
3.95 0.45(4)
5.75
3.95 0.45(5)
0.45n 3.95
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Additional Example 4 Continued
A DVD costs 3.95 to rent, plus 0.45 for each
day it is returned late. Find a function that
describes the arithmetic sequence, and then find
the total cost of renting the DVD and returning
it 9 days late.
0.45n 3.95
Write a function to find the 9th term.
0.45(9) 3.95
Substitute 9 for n.
4.05 3.95
Multiply.
8.00
Add.
It will cost 8.00 to rent a movie and return it
9 days late.
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Check It Out Example 4
A personal pizza with cheese costs 3.99 to buy,
plus 0.25 for each additional topping. Find a
function that describes the arithmetic sequence,
and then find the total cost of buying a personal
pizza with 7 additional toppings.
n 3.99 0.25n y
1
2
3
4
n
Multiply n by 0.25, and then add the 3.99 pizza
cost.
3.99 0.25(1)
4.24
3.99 0.25(2)
4.49
3.99 0.25(3)
4.74
3.99 0.25(4)
4.99
3.99 0.25(5)
0.25n 3.99
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Check It Out Example 4 Continued
A personal pizza with cheese costs 3.99 to buy,
plus 0.25 for each additional topping. Find a
function that describes the arithmetic sequence,
and then find the total cost of buying a personal
pizza with 7 additional toppings.
0.25n 3.99
Write a function to find the 7th term.
0.25(7) 3.99
Substitute 7 for n.
1.75 3.99
Multiply.
5.74
Add.
It will cost 5.74 to buy a personal pizza with 7
additional toppings.
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Lesson Quiz Part 1
Find the common difference in each arithmetic
sequence.
1. 4, 2, 0, 2,
2
4 3
8 3
10 3
2 3
2. , 2, , ,
Find the next three terms in each arithmetic
sequence.
2, 7, 12
3. 18, 13, 8, 3, 4. 3.6, 5, 6.4, 7.8,
9.2, 10.6, 12
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Lesson Quiz Part 2
Find a function that describes the arithmetic
sequence.
5. 5, 10, 15, 20,
Possible Answer y 5n
6. 1, 2, 5, 8,
Possible Answer y 3n 4
7. A runner finishes a lap in 55 seconds. Her
goal is to decrease her time by two seconds every
week. Find a function that describes the
arithmetic sequence and find how many seconds her
lap should be after 12 weeks of training.
Possible Answer s -2w 55 31 seconds