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13.2 Arithmetic Sequences

A constant common difference between terms, recursive and explicit nth-term formulas, and linear models like steady salary raises.

Algebra and Trigonometry 2e, 13.2 Arithmetic Sequences

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50 exercises

13.2 Exercise 1

Determine if each sequence is arithmetic. If so, indicate the common difference.

ⓐ 5,9,13,17,21,25,…5 , 9 , 13 , 17 , 21 , 25 , \ldots

ⓑ 4,9,12,17,20,25,…4 , 9 , 12 , 17 , 20 , 25 , \ldots

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13.2 Exercise 4

Write the first five terms of the sequence where the first term is 5 and the common difference is d=−6.d = −6 .

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13.2 Exercise 6

Find the general term (nth term) of an arithmetic sequence.

  1. ⓐ Find the twenty-first term of a sequence where the first term is three and the common difference is eight.
  2. ⓑ Find the eleventh term of a sequence where the third term is 19 and the common difference is five. Give the formula for the general term.
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13.2 Exercise 10

Is the given sequence arithmetic? If so, find the common difference.

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13.2 Exercise 11

Is the given sequence arithmetic? If so, find the common difference.

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Writing Terms of Arithmetic Sequences

Write the first five terms of the arithmetic sequence with a 1 =17 a_{1} = 17 and d=−3 d = - 3 .

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13.2 Exercise 13

List the first five terms of the arithmetic sequence with a 1 =1 a_{1} = 1 and d=5 d = 5 .

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Writing Terms of Arithmetic Sequences

Given a 1 =8 a_{1} = 8 and a 4 =14 a_{4} = 14 , find a 5 a_{5} .

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13.2 Exercise 15

Given a 3 =7 a_{3} = 7 and a 5 =17 a_{5} = 17 , find a 2 a_{2} .

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Writing a Recursive Formula for an Arithmetic Sequence

Write a recursive formula for the arithmetic sequence.

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13.2 Exercise 17

Write a recursive formula for the arithmetic sequence.

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Writing the n th Term Explicit Formula for an Arithmetic Sequence

Write an explicit formula for the arithmetic sequence.

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13.2 Exercise 19

Write an explicit formula for the following arithmetic sequence.

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Finding the Number of Terms in a Finite Arithmetic Sequence

Find the number of terms in the finite arithmetic sequence.

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13.2 Exercise 21

Find the number of terms in the finite arithmetic sequence.

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Solving Application Problems with Arithmetic Sequences

A five-year old child receives an allowance of $1 each week. His parents promise him an annual increase of $2 per week.

  1. ⓐWrite a formula for the child’s weekly allowance in a given year.
  2. ⓑWhat will the child’s allowance be when he is 16 years old?
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13.2 Exercise 23

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?

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13.2 Exercise 24

What is an arithmetic sequence?

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13.2 Exercise 26

How do we determine whether a sequence is arithmetic?

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13.2 Exercise 28

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

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13.2 Exercise 30

Find the common difference for the arithmetic sequence provided.

{ 0 , 1 2 , 1 , 3 2 , 2 , ... } \left\{0 , \frac{1}{2} , 1 , \frac{3}{2} , 2 , ...\right\}

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13.2 Exercise 32

Determine whether the sequence is arithmetic. If so find the common difference.

{ 4 , 16 , 64 , 256 , 1024 , ... } \left\{4 , 16 , 64 , 256 , 1024 , ...\right\}

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13.2 Exercise 34

Write the first five terms of the arithmetic sequence given the first term and common difference.

a 1 =0 a_{1} = 0 , d= 2 3 d = \frac{2}{3}

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13.2 Exercise 36

Write the first five terms of the arithmetic series given two terms.

a 13 =−60, a 33 =−160 a_{13} = - 60 , a_{33} = - 160

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13.2 Exercise 38

Find the specified term for the arithmetic sequence given the first term and common difference.

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

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13.2 Exercise 40

Find the specified term for the arithmetic sequence given the first term and common difference.

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

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13.2 Exercise 42

Find the first term given two terms from an arithmetic sequence.

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.

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13.2 Exercise 44

Find the first term given two terms from an arithmetic sequence.

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.

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13.2 Exercise 46

Find the first term given two terms from an arithmetic sequence.

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.

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13.2 Exercise 48

Find the specified term given two terms from an arithmetic sequence.

a 3 =−17.1 a_{3} = - 17.1 and a 10 =−15.7. a_{10} = - 15.7. Find a 21 . a_{21} .

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13.2 Exercise 50

Use the recursive formula to write the first five terms of the arithmetic sequence.

a 1 =−19; a n = a n−1 −1.4 a_{1} = - 19 ; a_{n} = a_{n - 1} - 1.4

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13.2 Exercise 52

Write a recursive formula for each arithmetic sequence.

a ={17,26,35,...} a = \left\{17 , 26 , 35 , ...\right\}

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13.2 Exercise 54

Write a recursive formula for each arithmetic sequence.

a ={12,17,22,...} a = \left\{12 , 17 , 22 , ...\right\}

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13.2 Exercise 56

Write a recursive formula for each arithmetic sequence.

a ={8.9,10.3,11.7,...} a = \left\{8.9 , 10.3 , 11.7 , ...\right\}

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13.2 Exercise 58

Write a recursive formula for each arithmetic sequence.

a ={ 1 5 , 9 20 , 7 10 ,... } a = \left\{\frac{1}{5} , \frac{9}{20} , \frac{7}{10} , ...\right\}

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13.2 Exercise 60

Write a recursive formula for each arithmetic sequence.

a ={ 1 6 ,− 11 12 ,−2,... } a = \left\{\frac{1}{6} , - \frac{11}{12} , - 2 , ...\right\}

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13.2 Exercise 62

Write a recursive formula for the given arithmetic sequence, and then find the specified term.

a ={4, 11, 18, ...}; a = \left\{4 ,\textrm{ } 11 ,\textrm{ } 18 ,\textrm{ } ...\right\} ; Find the 14th term.

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13.2 Exercise 64

Use the explicit formula to write the first five terms of the arithmetic sequence.

a n =24−4n a_{n} = 24 - 4 n

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13.2 Exercise 66

Write an explicit formula for each arithmetic sequence.

a ={3,5,7,...} a = \left\{3 , 5 , 7 , ...\right\}

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13.2 Exercise 68

Write an explicit formula for each arithmetic sequence.

a ={−5, 95, 195, ...} a = \left\{- 5 ,\textrm{ } 95 ,\textrm{ } 195 ,\textrm{ } ...\right\}

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13.2 Exercise 70

Write an explicit formula for each arithmetic sequence.

a ={1.8, 3.6, 5.4, ...} a = \left\{1.8 ,\textrm{ } 3.6 ,\textrm{ } 5.4 ,\textrm{ } ...\right\}

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13.2 Exercise 72

Write an explicit formula for each arithmetic sequence.

a ={15.8,18.5,21.2,...} a = \left\{15.8 , 18.5 , 21.2 , ...\right\}

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13.2 Exercise 74

Write an explicit formula for each arithmetic sequence.

a ={ 0, 1 3 , 2 3 ,... } a = \left\{0 , \frac{1}{3} , \frac{2}{3} , ...\right\}

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13.2 Exercise 76

Find the number of terms in the given finite arithmetic sequence.

a ={3,−4,−11, ...,−60} a = \left\{3 , - 4 , - 11 ,\textrm{ } ... , - 60\right\}

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13.2 Exercise 78

Find the number of terms in the given finite arithmetic sequence.

a ={ 1 2 ,2, 7 2 ,...,8 } a = \left\{\frac{1}{2} , 2 , \frac{7}{2} , ... , 8\right\}

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13.2 Exercise 84

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

What are the f…

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13.2 Exercise 90

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

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13.2 Exercise 92

Find the 11th term of the arithmetic sequence {3a−2b,a+2b,−a+6b…}. \left\{3 a - 2 b , a + 2 b , - a + 6 b \ldots\right\} .

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13.2 Exercise 94

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

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13.2 Exercise 96

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

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Adapted from College Algebra © 2021 Rice University. Licensed under CC BY 4.0. Sources: College Algebra 2e, Algebra and Trigonometry 2e, Precalculus 2e, College Algebra with Corequisite Support 2e. Imported at revision 4922e46.