Chapter 8

Sequences, Induction, and Probability

8.2 Arithmetic Sequences

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Objectives:

• • • • Find the common difference for an arithmetic sequence.

Write terms of an arithmetic sequence.

Use the formula for the general terms of an arithmetic sequence.

Use the formula for the sum of the first

n

terms of an arithmetic sequence.

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Definition of an Arithmetic Sequence

An

arithmetic sequence

is a sequence in which each term after the first differs from the preceding term by a constant amount. The difference between consecutive terms is called the

common difference

of the sequence.

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Example: Writing the Terms of an Arithmetic Sequence

Write the first six terms of the arithmetic sequence in which

a

1 = 100 and

a n

=

a n

–1 – 30.

a a a

3

a a

5

a

1 2 4 6  100   1 

a

2  

a

3  

a

4  

a

5  30 30 30       70 40 20   50 The terms are 100, 70, 40, 10, – 20, – 50.

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General Term of an Arithmetic Sequence

The

n

th term (the general term) of an arithmetic sequence with first term

a

1 and common difference

d

is

a n

  1 (

n

 1)

d

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Example: Using the Formula for the General Term of an Arithmetic Sequence

Find the ninth term of the arithmetic sequence whose first term is 6 and whose common difference is –5.

To find the ninth term,

a

9 , we replace

n

with 9,

a a n

9

a

1 with 6, and

a a

1 1 (

n

 1) (9 1)

d d d

with –5.

    6 8( 5) in the formula 40   34 The ninth term is –34.

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Example: Using an Arithmetic Sequence to Model Changes in the U.S. Population

The data in the graph show that in 2010, 16% of the U.S. population was Latino. On average, this is projected to increase by approximately 0.35% per year.

Write a formula for the

n

th term of the arithmetic sequence that describes the percentage of the U.S.

population that will be Latino

n

years after 2009.

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Example: Using an Arithmetic Sequence to Model Changes in the U.S. Population (continued)

a n

  1 (

n

 1)

d a

1  16

a d n

 0.35

  1 (

n

 1)

d

 

n

 1) 

n

 0.35

 0.35

n

 15.65

The formula for the

n

th term of the arithmetic sequence that describes the percentage of the U.S. population that will be Latino

n

years after 2009 is

a n

 0.35

n

 15.65.

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Example: Using an Arithmetic Sequence to Model Changes in the U.S. Population (continued)

We have found that the formula for the

n

th term of the arithmetic sequence that describes the percentage of the U.S. population that will be Latino

n a n

 0.35

years after 2009 is

n

 15.65.

What percentage of the U.S. population is projected to be Latino in 2030?

a

21 

n

  23  21 23% of the U.S. population is projected to be Latino in 2030.

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The Sum of the First n Terms of an Arithmetic Sequence

The sum,

S n

,of the first sequence is given by

n

terms of an arithmetic

S n



n

2 (

a

1 

a n

),

in which

a

1 is the first term and

a n

is the

n

th term.

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Example: Finding the Sum of n Terms of an Arithmetic Sequence

Find the sum of the first 15 terms of the arithmetic sequence: 3, 6, 9, 12, … We use the formula for the general term of a sequence

S

to find

a

15 . The common difference is 3.

a a

15

n

  1 (

n

 1)

d

 45 The sum of the first 15 terms of the

S n



n

2 (

a

1 

a n

),

sequence is 360.

15 

15 2



15 (48) 2



360

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