Learn to find terms in an arithmetic sequence'

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Learn to find terms in an arithmetic sequence.
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Vocabulary
sequence term arithmetic sequence common
difference
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A sequence is a list of numbers or objects,
called terms, in a certain order. 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. A. 5, 8, 11, 14,
17, . . .
Find the difference of each term and the term
before it.
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. B. 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. C. 65, 60, 55,
50, 45, . . .
Find the difference of each term and the term
before it.
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 1D Identifying Arithmetic
Sequences
Determine if the sequence could be arithmetic. If
so, give the common difference. D. 5.7, 5.8, 5.9,
6, 6.1, . . .
Find the difference of each term and the term
before it.
5.7 5.8 5.9 6 6.1, . . .
0.1
0.1
0.1
0.1
The sequence could be arithmetic with a common
difference of 0.1.
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Additional Example 1E Identifying Arithmetic
Sequences
Determine if the sequence could be arithmetic. If
so, give the common difference. E. 1, 0, -1, 0,
1, . . .
Find the difference of each term and the term
before it.
1 0 1 0 1, . . .
1
1
1
1
The sequence is not arithmetic.
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Try This Example 1A
Determine if the sequence could be arithmetic. If
so, give the common difference. A. 1, 2, 3, 4, 5,
. . .
Find the difference of each term and the term
before it.
1 2 3 4 5, . . .
1
1
1
1
The sequence could be arithmetic with a common
difference of 1.
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Try This Example 1B
Determine if the sequence could be arithmetic. If
so, give the common difference. B. 1, 3, 7, 8,
12,
Find the difference of each term and the term
before it.
1 3 7 8 12, . . .
4
1
4
2
The sequence is not arithmetic.
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Try This Example 1C
Determine if the sequence could be arithmetic. If
so, give the common difference. C. 11, 22, 33,
44, 55, . . .
Find the difference of each term and the term
before it.
11 22 33 44 55, . . .
11
11
11
11
The sequence could be arithmetic with a common
difference of 11.
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Try This Example 1D
Determine if the sequence could be arithmetic. If
so, give the common difference. D. 1, 1, 1, 1, 1,
1, . . .
Find the difference of each term and the term
before it.
1 1 1 1 1, . . .
0
0
0
0
The sequence could be arithmetic with a common
difference of 0.
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Try This Example 1E
Determine if the sequence could be arithmetic. If
so, give the common difference. E. 2, 4, 6, 8, 9,
. . .
Find the difference of each term and the term
before it.
2 4 6 8 9, . . .
1
2
2
2
The sequence is not arithmetic.
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Additional Example 2A Finding a Given Term of an
Arithmetic Sequence
Find the given term in the arithmetic
sequence. A. 10th term 1, 3, 5, 7, . . .
an a1 (n 1)d
a10 1 (10 1)2
a10 19
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Additional Example 2B Finding a Given Term of an
Arithmetic Sequence
Find the given term in the arithmetic
sequence. B. 18th term 100, 93, 86, 79, . . .
an a1 (n 1)d
a18 100 (18 1)(7)
a18 -19
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Additional Example 2C Finding a Given Term of an
Arithmetic Sequence
Find the given term in the arithmetic
sequence. C. 21st term 25, 25.5, 26, 26.5, . . .
an a1 (n 1)d
a21 25 (21 1)(0.5)
a21 35
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Additional Example 2D Finding a Given Term of an
Arithmetic Sequence
Find the given term in the arithmetic
sequence. D. 14th term a1 13, d 5
an a1 (n 1)d
a14 13 (14 1)5
a14 78
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Try This Example 2A
Find the given term in the arithmetic
sequence. A. 15th term 1, 3, 5, 7, . . .
an a1 (n 1)d
a15 1 (15 1)2
a15 29
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Try This Example 2B
Find the given term in the arithmetic
sequence. B. 50th term 100, 93, 86, 79, . . .
an a1 (n 1)d
a50 100 (50 1)(-7)
a50 243
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Try This Example 2C
Find the given term in the arithmetic
sequence. C. 41st term 25, 25.5, 26, 26.5, . . .
an a1 (n 1)d
a41 25 (41 1)(0.5)
a41 45
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Try This Example 2D
Find the given term in the arithmetic
sequence. D. 2nd term a1 13, d 5
an a1 (n 1)d
a2 13 (2 1)5
a2 18
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You can use the formula for the nth term of an
arithmetic sequence to solve for other variables.
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Additional Example 3 Application
The senior class held a bake sale. At the
beginning of the sale, there was 20 in the cash
box. Each item in the sale cost 50 cents. At the
end of the sale, there was 63.50 in the cash
box. How many items were sold during the bake
sale?
Identify the arithmetic sequence
20.5, 21, 21.5, 22, . . .
a1 20.5
Let a1 20.5 money after first sale.
d 0.5
an 63.5
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Additional Example 3 Continued
Let n represent the item number in which the cash
box will contain 63.50. Use the formula for
arithmetic sequences.
an a1 (n 1) d
Solve for n.
63.5 20.5 (n 1)(0.5)
Distributive Property.
63.5 20.5 0.5n 0.5
63.5 20 0.5n
Combine like terms.
Subtract 20 from both sides.
43.5 0.5n
Divide both sides by 0.5.
87 n
During the bake sale, 87 items are sold in order
for the cash box to contain 63.50.
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Try This Example 3
Johnnie is selling pencils for student council.
At the beginning of the day, there was 10 in his
money bag. Each pencil costs 25 cents. At the end
of the day, he had 40 in his money bag. How many
pencils were sold during the day?
Identify the arithmetic sequence 10.25, 10.5,
10.75, 11,
a1 10.25
Let a1 10.25 money after first sale.
d 0.25
an 40
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Try This Example 3 Continued
Let n represent the number of pencils in which he
will have 40 in his money bag. Use the formula
for arithmetic sequences.
an a1 (n 1)d
40 10.25 (n 1)(0.25)
Solve for n.
40 10.25 0.25n 0.25
Distributive Property.
Combine like terms.
40 10 0.25n
Subtract 10 from both sides.
30 0.25n
120 n
Divide both sides by 0.25.
120 pencils are sold in order for his money bag
to contain 40.
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Lesson Quiz
Determine if each sequence could be arithmetic.
If so, give the common difference. 1. 42, 49, 56,
63, 70, . . . 2. 1, 2, 4, 8, 16, 32, . .
. Find the given term in each arithmetic
sequence. 3. 15th term a1 7, d 5 4. 24th
term 1, , , , 2 5. 52nd term a1 14.2 d
1.2
yes 7
no
77
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