Pre-Algebra 12-1 Arithmetic Sequences Pre-Algebra HOMEWORK Page 606 #1-9

Pre-Algebra

12-1 Arithmetic Sequences

Pre-Algebra HOMEWORK

Page 606

#1-9

Pre-Algebra


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


Today’s Learning Goal Assignment

Learn to find terms in an arithmetic sequence.

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12-1 Arithmetic Sequences12-1 Arithmetic Sequences

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Warm UpWarm Up

Problem of the DayProblem of the Day

Lesson PresentationLesson Presentation

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Warm UpFind the next two numbers in the pattern, using the simplest rule you can find.

1. 1, 5, 9, 13, . . .

2. 100, 50, 25, 12.5, . . .

3. 80, 87, 94, 101, . . .

4. 3, 9, 7, 13, 11, . . .

17, 21

6.25, 3.125

108, 115

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17, 15

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Problem of the Day

Write the last part of this set of equations so that its graph is the letter W.y = –2x + 4 for 0 x 2y = 2x – 4 for 2 < x 4y = –2x + 12 for 4 < x 6

Possible answer: y = 2x – 12 for 6 < x 8

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Vocabulary

sequencetermarithmetic sequencecommon 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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Determine if the sequence could be arithmetic. If so, give the common difference.

A. 5, 8, 11, 14, 17, . . .

Additional Example 1A: Identifying Arithmetic Sequences

Find the difference of each term and the term before it.

The sequence could be arithmetic with a common difference of 3.

5 8 11 14 17, . . .

3333

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A. 1, 2, 3, 4, 5, . . .

Try This: Example 1A



1 2 3 4 5, . . .

1111

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B. 1, 3, 6, 10, 15, . . .

Additional Example 1B: Identifying Arithmetic Sequences

The sequence is not arithmetic.


1 3 6 10 15, . . .

5432

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B. 1, 3, 7, 8, 12, …

Try This: Example 1B



1 3 7 8 12, . . .

4142

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C. 65, 60, 55, 50, 45, . . .

Additional Example 1C: Identifying Arithmetic Sequences

The sequence could be arithmetic with a common difference of –5.


65 60 55 50 45, . . .

–5–5–5–5

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C. 11, 22, 33, 44, 55, . . .

Try This: Example 1C



11 22 33 44 55, . . .

11111111

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D. 5.7, 5.8, 5.9, 6, 6.1, . . .

Additional Example 1D: Identifying Arithmetic Sequences

The sequence could be arithmetic with a common difference of 0.1.


5.7 5.8 5.9 6 6.1, . . .

0.10.10.10.1

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D. 1, 1, 1, 1, 1, 1, . . .

Try This: Example 1D



1 1 1 1 1, . . .

0000

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E. 1, 0, -1, 0, 1, . . .

Additional Example 1E: Identifying Arithmetic Sequences



1 0 –1 0 1, . . .

11–1–1

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E. 2, 4, 6, 8, 9, . . .

Try This: Example 1E



2 4 6 8 9, . . .

1222

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Writing Math

Subscripts are used to show the positions of terms in the sequence. The first term is a1, the second is a2, and so on.

FINDING THE nth TERM OF AN ARITHMETIC SEQUENCE

The nth term an of an arithmetic sequence with common difference d is

an = a1 + (n – 1)d.

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Find the given term in the arithmetic sequence.

A. 10th term: 1, 3, 5, 7, . . .

Additional Example 2A: Finding a Given Term of an Arithmetic Sequence

an = a1 + (n – 1)d

a10 = 1 + (10 – 1)2

a10 = 19

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A. 15th term: 1, 3, 5, 7, . . .

Try This: Example 2A

an = a1 + (n – 1)d

a15 = 1 + (15 – 1)2

a15 = 29

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B. 18th term: 100, 93, 86, 79, . . .

Additional Example 2B: Finding a Given Term of an Arithmetic Sequence

an = a1 + (n – 1)d

a18 = 100 + (18 – 1)(–7)

a18 = -19

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B. 50th term: 100, 93, 86, 79, . . .

Try This: Example 2B

an = a1 + (n – 1)d

a50 = 100 + (50 – 1)(-7)

a50 = –243

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C. 21st term: 25, 25.5, 26, 26.5, . . .

Additional Example 2C: Finding a Given Term of an Arithmetic Sequence

an = a1 + (n – 1)d

a21 = 25 + (21 – 1)(0.5)

a21 = 35

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C. 41st term: 25, 25.5, 26, 26.5, . . .

Try This: Example 2C

an = a1 + (n – 1)d

a41 = 25 + (41 – 1)(0.5)

a41 = 45

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D. 14th term: a1 = 13, d = 5

Additional Example 2D: Finding a Given Term of an Arithmetic Sequence

an = a1 + (n – 1)d

a14 = 13 + (14 – 1)5

a14 = 78

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D. 2nd term: a1 = 13, d = 5

Try This: Example 2D

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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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?

Additional Example 3: Application

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)

63.5 = 20.5 + 0.5n – 0.5 Distributive Property.

63.5 = 20 + 0.5n Combine like terms.

87 = n

Subtract 20 from both sides.

During the bake sale, 87 items are sold in order for the cash box to contain $63.50.

43.5 = 0.5n

Divide both sides by 0.5.

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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?

Try This: Example 3

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 ContinuedLet 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

Solve for n.40 = 10.25 + (n – 1)(0.25)

40 = 10.25 + 0.25n – 0.25 Distributive Property.

40 = 10 + 0.25n Combine like terms.

120 = n

Subtract 10 from both sides.

120 pencils are sold in order for his money bag to contain $40.

30 = 0.25n

Divide both sides by 0.25.

Pre-Algebra


Lesson QuizDetermine 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

no

yes; 7

77

54

32

74

, or 6.7527 4

–47

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Pre-Algebra 12-1 Arithmetic Sequences Pre-Algebra HOMEWORK Page 606 #1-9