Chapter 10 Further Topics in Algebra

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SECTION 10.2

Arithmetic Sequences; Partial Sums

OBJECTIVES 1 2 Identify an arithmetic sequence and find its common difference.

Find the sum of the first arithmetic sequence.

n

terms of an © 2010 Pearson Education, Inc. All rights reserved

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DEFINITION OF AN ARITHMETIC SEQUENCE The sequence

a

1 ,

a

2 ,

a

3 ,

a

4 , … ,

a n , …

is an

arithmetic sequence

, or an

arithmetic progression

d

if there is a number each term in the sequence except the first is obtained from the preceding term by adding to it. The number

d d

is called the such that

common difference

of the arithmetic sequence. We have

d

=

a n

+ 1 –

a n

,

n

≥ 1.

3

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RECURSIVE DEFINITION OF AN ARITHMETIC SEQUENCE An arithmetic sequence formula

a a

1

n

,

a

+ 1 2 , =

a a

3

n

,

a

+ 4 , … ,

d a

for

n n , …

can be defined recursively. The recursive ≥ 1 defines an arithmetic sequence with first term

a

1 and common difference

d

.

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nTH TERM OF AN ARITHMETIC SEQUENCE If a sequence

a

1 ,

a

2 ,

a

3 ,

…

is an arithmetic sequence, then its

n

th term,

a n

, is given by

a n

=

a

1 + (

n

– 1)

d

, where

a

1 is the first term and common difference.

d

is the © 2010 Pearson Education, Inc. All rights reserved

5

Practice Problem © 2010 Pearson Education, Inc. All rights reserved

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Practice Problem © 2010 Pearson Education, Inc. All rights reserved

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EXAMPLE 3 Finding the Common Difference of an Arithmetic Sequence Find the common difference

d

and the

n

th term

a n

of an arithmetic sequence whose 5th term is 15 and whose 20th term is 45.

Solution

a n

4 5   1   1 

n

 20  1   1 

d d

45   1 19

d a n

15   1

a

1  

n

 1 

d

1 

d

15   1 4

d

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EXAMPLE 3 Finding the Common Difference of an Arithmetic Sequence Solution continued Solving the system of equations 45 15  

a

1

a

1   19 4

d d

gives

a

1 = 7 and

d a n a n

= 2.

  1  

n n

  1  1  2

d a n

2

n

2

n

 5 The

n

th term is given by

a n

= 2

n

+ 5,

n

≥ 1.

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Practice Problem © 2010 Pearson Education, Inc. All rights reserved

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SUM OF n TERMS OF AN ARITHMETIC SEQUENCE Let

d a

1 ,

a

2 ,

a

arithmetic sequence with common difference . The sum 3

S

,

n … a n

be the first of these

n n

terms of an terms is given by

S n



a

1  

a n

2   , where

a n

=

a

1 + (

n

– 1)

d

.

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EXAMPLE 4 Finding the Sum of Terms of a Finite Arithmetic Sequence Find the sum of the arithmetic sequence of numbers: 1 + 4 + 7 + … + 25 Solution Arithmetic sequence with

a

1 = 1 and First find the number of terms.

d a n

2 5 24   1  

n n

  1 1  3 

d

 

n

 1  3 8 

n



n

 1 9 = 3. © 2010 Pearson Education, Inc. All rights reserved

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EXAMPLE 4 Finding the Sum of Terms of a Finite Arithmetic Sequence Solution continued

S n S

9   

a

1 

a n

2     1  25    2     117 Thus 1 + 4 + 7 + … + 25 = 117.

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Practice Problem © 2010 Pearson Education, Inc. All rights reserved

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Practice Problem © 2010 Pearson Education, Inc. All rights reserved

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