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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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