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# Arithmetic Sequences

Module by: First Last. E-mail the author

Summary: In this section, you will:

• Find the common difference for an arithmetic sequence.
• Write terms of an arithmetic sequence.
• Use a recursive formula for an arithmetic sequence.
• Use an explicit formula for an arithmetic sequence.

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Companies often make large purchases, such as computers and vehicles, for business use. The book-value of these supplies decreases each year for tax purposes. This decrease in value is called depreciation. One method of calculating depreciation is straight-line depreciation, in which the value of the asset decreases by the same amount each year.

#### Try It:

##### Exercise 8

A woman decides to go for a 10-minute run every day this week and plans to increase the time of her daily run by 4 minutes each week. Write a formula for the time of her run after n weeks. How long will her daily run be 8 weeks from today?

###### Solution

The formula is T n =10+4n, T n =10+4n, and it will take her 42 minutes.

#### Media:

Access this online resource for additional instruction and practice with arithmetic sequences.

## Key Equations

 recursive formula for nth term of an arithmetic sequence a n = a n−1 +d n≥2 a n = a n−1 +d n≥2 explicit formula for nth term of an arithmetic sequence a n = a 1 +d(n−1) a n = a 1 +d(n−1)

## Key Concepts

• An arithmetic sequence is a sequence where the difference between any two consecutive terms is a constant.
• The constant between two consecutive terms is called the common difference.
• The common difference is the number added to any one term of an arithmetic sequence that generates the subsequent term. See Example 1.
• The terms of an arithmetic sequence can be found by beginning with the initial term and adding the common difference repeatedly. See Example 2 and Example 3.
• A recursive formula for an arithmetic sequence with common difference d d is given by a n = a n1 +d,n2. a n = a n1 +d,n2. See Example 4.
• As with any recursive formula, the initial term of the sequence must be given.
• An explicit formula for an arithmetic sequence with common difference d d is given by a n = a 1 +d(n1). a n = a 1 +d(n1). See Example 5.
• An explicit formula can be used to find the number of terms in a sequence. See Example 6.
• In application problems, we sometimes alter the explicit formula slightly to a n = a 0 +dn. a n = a 0 +dn. See Example 7.

## Section Exercises

### Verbal

#### Exercise 9

What is an arithmetic sequence?

##### Solution

A sequence where each successive term of the sequence increases (or decreases) by a constant value.

#### Exercise 10

How is the common difference of an arithmetic sequence found?

#### Exercise 11

How do we determine whether a sequence is arithmetic?

##### Solution

We find whether the difference between all consecutive terms is the same. This is the same as saying that the sequence has a common difference.

#### Exercise 12

What are the main differences between using a recursive formula and using an explicit formula to describe an arithmetic sequence?

#### Exercise 13

Describe how linear functions and arithmetic sequences are similar. How are they different?

##### Solution

Both arithmetic sequences and linear functions have a constant rate of change. They are different because their domains are not the same; linear functions are defined for all real numbers, and arithmetic sequences are defined for natural numbers or a subset of the natural numbers.

### Algebraic

For the following exercises, find the common difference for the arithmetic sequence provided.

#### Exercise 14

{ 5 , 11 , 17 , 23 , 29 , ... } { 5 , 11 , 17 , 23 , 29 , ... }

#### Exercise 15

{ 0 , 1 2 , 1 , 3 2 , 2 , ... } { 0 , 1 2 , 1 , 3 2 , 2 , ... }

##### Solution

The common difference is 1 2 1 2

For the following exercises, determine whether the sequence is arithmetic. If so find the common difference.

#### Exercise 16

{ 11.4 , 9.3 , 7.2 , 5.1 , 3 , ... } { 11.4 , 9.3 , 7.2 , 5.1 , 3 , ... }

#### Exercise 17

{ 4 , 16 , 64 , 256 , 1024 , ... } { 4 , 16 , 64 , 256 , 1024 , ... }

##### Solution

The sequence is not arithmetic because 1646416. 1646416.

For the following exercises, write the first five terms of the arithmetic sequence given the first term and common difference.

#### Exercise 18

a 1 =−25 a 1 =−25 , d=−9 d=−9

#### Exercise 19

a 1 =0 a 1 =0 , d= 2 3 d= 2 3

##### Solution

0, 2 3 , 4 3 ,2, 8 3 0, 2 3 , 4 3 ,2, 8 3

For the following exercises, write the first five terms of the arithmetic series given two terms.

#### Exercise 20

a 1 =17, a 7 =31 a 1 =17, a 7 =31

#### Exercise 21

a 13 =60, a 33 =160 a 13 =60, a 33 =160

##### Solution

0 , 5 , 10 , 15 , 20 0 , 5 , 10 , 15 , 20

For the following exercises, find the specified term for the arithmetic sequence given the first term and common difference.

#### Exercise 22

First term is 3, common difference is 4, find the 5th term.

#### Exercise 23

First term is 4, common difference is 5, find the 4th term.

a 4 =19 a 4 =19

#### Exercise 24

First term is 5, common difference is 6, find the 8th term.

#### Exercise 25

First term is 6, common difference is 7, find the 6th term.

a 6 =41 a 6 =41

#### Exercise 26

First term is 7, common difference is 8, find the 7th term.

For the following exercises, find the first term given two terms from an arithmetic sequence.

#### Exercise 27

Find the first term or a 1 a 1 of an arithmetic sequence if a 6 =12 a 6 =12 and a 14 =28. a 14 =28.

a 1 =2 a 1 =2

#### Exercise 28

Find the first term or a 1 a 1 of an arithmetic sequence if a 7 =21 a 7 =21 and a 15 =42. a 15 =42.

#### Exercise 29

Find the first term or a 1 a 1 of an arithmetic sequence if a 8 =40 a 8 =40 and a 23 =115. a 23 =115.

a 1 =5 a 1 =5

#### Exercise 30

Find the first term or a 1 a 1 of an arithmetic sequence if a 9 =54 a 9 =54 and a 17 =102. a 17 =102.

#### Exercise 31

Find the first term or a 1 a 1 of an arithmetic sequence if a 11 =11 a 11 =11 and a 21 =16. a 21 =16.

##### Solution

a 1 =6 a 1 =6

For the following exercises, find the specified term given two terms from an arithmetic sequence.

#### Exercise 32

a 1 =33 a 1 =33 and a 7 =15. a 7 =15. Find a 4 . a 4 .

#### Exercise 33

a 3 =17.1 a 3 =17.1 and a 10 =15.7. a 10 =15.7. Find a 21 . a 21 .

##### Solution

a 21 =13.5 a 21 =13.5

For the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence.

#### Exercise 34

a 1 =39;  a n = a n1 3 a 1 =39;  a n = a n1 3

#### Exercise 35

a 1 =19;  a n = a n1 1.4 a 1 =19;  a n = a n1 1.4

##### Solution

19,20.4,21.8,23.2,24.6 19,20.4,21.8,23.2,24.6

For the following exercises, write a recursive formula for each arithmetic sequence.

#### Exercise 36

a n ={ 40,60,80,... } a n ={ 40,60,80,... }

#### Exercise 37

a n ={17,26,35,...} a n ={17,26,35,...}

##### Solution

a 1 =17;  a n = a n1 +9 n2 a 1 =17;  a n = a n1 +9 n2

#### Exercise 38

a n ={1,2,5,...} a n ={1,2,5,...}

#### Exercise 39

a n ={12,17,22,...} a n ={12,17,22,...}

##### Solution

a 1 =12;  a n = a n1 +5 n2 a 1 =12;  a n = a n1 +5 n2

#### Exercise 40

a n ={15,7,1,...} a n ={15,7,1,...}

#### Exercise 41

a n ={8.9,10.3,11.7,...} a n ={8.9,10.3,11.7,...}

##### Solution

a 1 =8.9;  a n = a n1 +1.4 n2 a 1 =8.9;  a n = a n1 +1.4 n2

#### Exercise 42

a n ={0.52,1.02,1.52,...} a n ={0.52,1.02,1.52,...}

#### Exercise 43

a n ={ 1 5 , 9 20 , 7 10 ,... } a n ={ 1 5 , 9 20 , 7 10 ,... }

##### Solution

a 1 = 1 5 ;  a n = a n1 + 1 4 n2 a 1 = 1 5 ;  a n = a n1 + 1 4 n2

#### Exercise 44

a n ={ 1 2 , 5 4 ,2,... } a n ={ 1 2 , 5 4 ,2,... }

#### Exercise 45

a n ={ 1 6 , 11 12 ,2,... } a n ={ 1 6 , 11 12 ,2,... }

##### Solution

1 = 1 6 ;  a n = a n1 13 12 n2 1 = 1 6 ;  a n = a n1 13 12 n2

For the following exercises, write a recursive formula for the given arithmetic sequence, and then find the specified term.

#### Exercise 46

a n ={741...}; a n ={741...}; Find the 17th term.

#### Exercise 47

a n ={41118...}; a n ={41118...}; Find the 14th term.

##### Solution

a 1 =4;  a n = a n1 +7;  a 14 =95 a 1 =4;  a n = a n1 +7;  a 14 =95

#### Exercise 48

a n ={2610...}; a n ={2610...}; Find the 12th term.

For the following exercises, use the explicit formula to write the first five terms of the arithmetic sequence.

#### Exercise 49

a n =244n a n =244n

##### Solution

First five terms: 20,16,12,8,4. 20,16,12,8,4.

#### Exercise 50

a n = 1 2 n 1 2 a n = 1 2 n 1 2

For the following exercises, write an explicit formula for each arithmetic sequence.

#### Exercise 51

a n ={3,5,7,...} a n ={3,5,7,...}

##### Solution

a n =1+2n a n =1+2n

#### Exercise 52

a n ={32,24,16,...} a n ={32,24,16,...}

#### Exercise 53

a n ={595195...} a n ={595195...}

##### Solution

a n =105+100n a n =105+100n

#### Exercise 54

a n ={−17−217−417,...} a n ={−17−217−417,...}

#### Exercise 55

a n ={1.83.65.4...} a n ={1.83.65.4...}

##### Solution

a n =1.8n a n =1.8n

#### Exercise 56

a n ={−18.1,−16.2,−14.3,...} a n ={−18.1,−16.2,−14.3,...}

#### Exercise 57

a n ={15.8,18.5,21.2,...} a n ={15.8,18.5,21.2,...}

##### Solution

a n =13.1+2.7n a n =13.1+2.7n

#### Exercise 58

a n ={ 1 3 , 4 3 ,−3... } a n ={ 1 3 , 4 3 ,−3... }

#### Exercise 59

a n ={ 0, 1 3 , 2 3 ,... } a n ={ 0, 1 3 , 2 3 ,... }

##### Solution

a n = 1 3 n 1 3 a n = 1 3 n 1 3

#### Exercise 60

a n ={ 5, 10 3 , 5 3 , } a n ={ 5, 10 3 , 5 3 , }

For the following exercises, find the number of terms in the given finite arithmetic sequence.

#### Exercise 61

a n ={3,4,11...,60} a n ={3,4,11...,60}

##### Solution

There are 10 terms in the sequence.

#### Exercise 62

a n ={1.2,1.4,1.6,...,3.8} a n ={1.2,1.4,1.6,...,3.8}

#### Exercise 63

a n ={ 1 2 ,2, 7 2 ,...,8 } a n ={ 1 2 ,2, 7 2 ,...,8 }

##### Solution

There are 6 terms in the sequence.

### Graphical

For the following exercises, determine whether the graph shown represents an arithmetic sequence.

#### Exercise 65

##### Solution

The graph does not represent an arithmetic sequence.

For the following exercises, use the information provided to graph the first 5 terms of the arithmetic sequence.

#### Exercise 66

a 1 =0,d=4 a 1 =0,d=4

#### Exercise 67

a 1 =9; a n = a n1 10 a 1 =9; a n = a n1 10

#### Exercise 68

a n =12+5n a n =12+5n

### Technology

For the following exercises, follow the steps to work with the arithmetic sequence a n =3n2 a n =3n2 using a graphing calculator:

• Press [MODE]
• Select SEQ in the fourth line
• Select DOT in the fifth line
• Press [ENTER]
• Press [Y=]
• nMin nMin is the first counting number for the sequence. Set nMin=1 nMin=1
• u(n) u(n) is the pattern for the sequence. Set u(n)=3n2 u(n)=3n2
• u(nMin) u(nMin) is the first number in the sequence. Set u(nMin)=1 u(nMin)=1
• Press [2ND] then [WINDOW] to go to TBLSET
• Set TblStart=1 TblStart=1
• Set ΔTbl=1 ΔTbl=1
• Set Indpnt: Auto and Depend: Auto
• Press [2ND] then [GRAPH] to go to the TABLE

#### Exercise 69

What are the first seven terms shown in the column with the heading u(n)? u(n)?

##### Solution

1,4,7,10,13,16,19 1,4,7,10,13,16,19

#### Exercise 70

Use the scroll-down arrow to scroll to n=50. n=50. What value is given for u(n)? u(n)?

#### Exercise 71

Press [WINDOW]. Set nMin=1,nMax=5,xMin=0,xMax=6,yMin=1, nMin=1,nMax=5,xMin=0,xMax=6,yMin=1, and yMax=14. yMax=14. Then press [GRAPH]. Graph the sequence as it appears on the graphing calculator.

##### Solution

For the following exercises, follow the steps given above to work with the arithmetic sequence a n = 1 2 n+5 a n = 1 2 n+5 using a graphing calculator.

#### Exercise 72

What are the first seven terms shown in the column with the heading u(n) u(n) in the TABLE feature?

#### Exercise 73

Graph the sequence as it appears on the graphing calculator. Be sure to adjust the WINDOW settings as needed.

### Extensions

#### Exercise 74

Give two examples of arithmetic sequences whose 4th terms are 9. 9.

#### Exercise 75

Give two examples of arithmetic sequences whose 10th terms are 206. 206.

##### Solution

Answers will vary. Examples: a n =20.6n a n =20.6n and a n =2+20.4n. a n =2+20.4n.

#### Exercise 76

Find the 5th term of the arithmetic sequence {9b,5b,b,}. {9b,5b,b,}.

#### Exercise 77

Find the 11th term of the arithmetic sequence {3a2b,a+2b,a+6b}. {3a2b,a+2b,a+6b}.

##### Solution

a 11 =17a+38b a 11 =17a+38b

#### Exercise 78

At which term does the sequence {5.4,14.5,23.6,...} {5.4,14.5,23.6,...} exceed 151?

#### Exercise 79

At which term does the sequence { 17 3 , 31 6 , 14 3 ,... } { 17 3 , 31 6 , 14 3 ,... } begin to have negative values?

##### Solution

The sequence begins to have negative values at the 13th term, a 13 = 1 3 a 13 = 1 3

#### Exercise 80

For which terms does the finite arithmetic sequence { 5 2 , 19 8 , 9 4 ,..., 1 8 } { 5 2 , 19 8 , 9 4 ,..., 1 8 } have integer values?

#### Exercise 81

Write an arithmetic sequence using a recursive formula. Show the first 4 terms, and then find the 31st term.

##### Solution

Answers will vary. Check to see that the sequence is arithmetic. Example: Recursive formula: a 1 =3, a n = a n1 3. a 1 =3, a n = a n1 3. First 4 terms: 3,0,3,6 a 31 =87 3,0,3,6 a 31 =87

#### Exercise 82

Write an arithmetic sequence using an explicit formula. Show the first 4 terms, and then find the 28th term.

## Glossary

arithmetic sequence:
a sequence in which the difference between any two consecutive terms is a constant
common difference:
the difference between any two consecutive terms in an arithmetic sequence

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