1212-1-1Arithmetic Sequences - Kyrene School District · 12-1 Arithmetic Sequences Determine if the...

Pre-Algebra

12-1 Arithmetic Sequences12-1 Arithmetic Sequences

Pre-Algebra

Warm Up

Problem of the Day

Lesson Presentation

Pre-Algebra

12-1 Arithmetic Sequences

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 2

y = 2x – 4 for 2 < x 4

y = –2x + 12 for 4 < x 6

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

Pre-Algebra


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

Additional Example 1E: Identifying Arithmetic

Sequences



1 0 –1 0 1, . . .

11–1–1

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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, 7, 8, 12, …

Try This: Example 1B



1 3 7 8 12, . . .

4142

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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. 1, 1, 1, 1, 1, 1, . . .

Try This: Example 1D



1 1 1 1 1, . . .

0000

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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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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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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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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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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. 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. 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. 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 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

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 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

no

yes; 7

77

54

32

74

, or 6.75274

–47

1212-1-1Arithmetic Sequences - Kyrene School District · 12-1 Arithmetic Sequences Determine if the...

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