12.1 Arithmetic Sequences Answer Key

Learning Objectives

By the end of this department, you will be able to:

• Determine if a sequence is arithmetic
• Find the general term ($n$thursday term) of an arithmetic sequence
• Find the sum of the first $n$ terms of an arithmetic sequence

Be Prepared 12.4

Before you become started, take this readiness quiz.

Evaluate $4n-1$ for the integers i, ii, iii, and 4.
If you missed this problem, review Case ane.6.

Be Prepared 12.five

Solve the system of equations: $\{\begin{array}{l}x+y=7\\ \mathrm{three}\mathrm{ten}+\mathrm{four}y=23\end{array}.$
If you missed this trouble, review Example iv.nine.

Be Prepared 12.6

If $f\left(n\right)=\frac{\mathrm{northward}}{2}\left(\mathrm{three}\mathrm{north}+5\right),$ find $f\left(1\right)+f\left(\mathrm{xx}\right).$
If you missed this trouble, review Instance 3.49.

Determine if a Sequence is Arithmetic

The terminal section introduced sequences and now nosotros will look at two specific types of sequences that each take special backdrop. In this department we will await at arithmetic sequences and in the next department, geometric sequences.

An arithmetics sequence is a sequence where the difference betwixt consecutive terms is constant. The difference betwixt consecutive terms in an arithmetic sequence, $a_{\mathrm{northward}}-a_{\mathrm{north}-1},$ is d, the common difference, for n greater than or equal to two.

Arithmetic Sequence

An arithmetics sequence is a sequence where the difference between consecutive terms is e'er the same.

The deviation between sequent terms, $a_{\mathrm{due\; north}}-a_{n-1},$ is d, the common difference, for due north greater than or equal to two.

In each of these sequences, the divergence between consecutive terms is constant, and then the sequence is arithmetic.

Case 12.13

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

ⓐ $5,\mathrm{ix},13,17,21,25\text{,}\text{…}$

ⓑ $\mathrm{four},9,12,17,20,25\text{,}\text{…}$

ⓒ $\mathrm{x},3,\mathrm{-4},\mathrm{-11},\mathrm{-18},\mathrm{-25}\text{,}\text{…}$

Endeavour It 12.25

Decide if each sequence is arithmetics. If then, betoken the mutual departure.

ⓐ $9,20,31,42,53,64\text{,}\text{…}$ ⓑ $12,\mathrm{vi},0,\mathrm{-6},\mathrm{-12},\mathrm{-18}\text{,}\text{…}$ ⓒ $7,\mathrm{ane},10,4,\mathrm{thirteen},7\text{,}\text{…}$

Try It 12.26

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

ⓐ $\mathrm{-4},4,2,10,8,16\text{,}\text{…}$ ⓑ $\mathrm{-3},\mathrm{-1},1,\mathrm{three},\mathrm{five},7\text{,}\text{…}$ ⓒ $7,\mathrm{two},\mathrm{-3},\mathrm{-8},\mathrm{-thirteen},\mathrm{-18}\text{,}\text{…}$

If we know the first term, $a_1,$ and the common difference, d, we tin list a finite number of terms of the sequence.

Example 12.xiv

Write the first 5 terms of the sequence where the start term is 5 and the common deviation is $d=\mathrm{-6}.$

Try It 12.27

Write the first five terms of the sequence where the outset term is 7 and the common difference is $d=\mathrm{-4}.$

Attempt It 12.28

Write the first five terms of the sequence where the first term is 11 and the mutual deviation is $d=\mathrm{-8}.$

Discover the General Term (due northth Term) of an Arithmetic Sequence

Just as we found a formula for the general term of a sequence, we can also find a formula for the general term of an arithmetic sequence.

Allow's write the first few terms of a sequence where the first term is $a_1$ and the mutual divergence is d. Nosotros will and so look for a pattern.

As we expect for a pattern we come across that each term starts with $a_1$ .

The first term adds 0d to the $a_1$ , the 2nd term adds id, the third term adds iid, the quaternary term adds 3d, and the 5th term adds 4d. The number of ds that were added to $a_{\mathrm{one}}$ is one less than the number of the term. This leads us to the post-obit

$$a_{\mathrm{due\; north}}=a_1+\left(n-1\right)d$$

Full general Term (norththursday term) of an Arithmetic Sequence

The general term of an arithmetic sequence with first term $a_1$ and the common difference d is

$$a_{\mathrm{due\; north}}=a_{\mathrm{i}}+\left(n-1\right)d$$

We will use this formula in the next example to find the 15^th term of a sequence.

Example 12.15

Find the fifteenth term of a sequence where the showtime term is 3 and the mutual difference is 6.

Endeavour Information technology 12.29

Notice the twenty-seventh term of a sequence where the first term is 7 and the common difference is ix.

Try It 12.30

Observe the eighteenth term of a sequence where the first term is 13 and the common divergence is $\mathrm{-vii}$ .

Sometimes we do not know the first term and we must apply other given data to find it before we find the requested term.

Instance 12.16

Find the twelfth term of a sequence where the seventh term is 10 and the common difference is $\mathrm{-2}$ . Give the formula for the full general term.

Try It 12.31

Find the eleventh term of a sequence where the 9th term is 8 and the common difference is $\mathrm{-3}.$ Requite the formula for the general term.

Endeavour It 12.32

Find the nineteenth term of a sequence where the fifth term is i and the mutual difference is $\mathrm{-4}.$ Requite the formula for the general term.

Sometimes the information given leads united states to ii equations in two unknowns. We and then utilise our methods for solving systems of equations to find the values needed.

Example 12.17

Find the starting time term and common difference of a sequence where the 5th term is xix and the eleventh term is 37. Give the formula for the general term.

Endeavor Information technology 12.33

Find the kickoff term and common difference of a sequence where the fourth term is 17 and the thirteenth term is 53. Give the formula for the general term.

Try Information technology 12.34

Detect the showtime term and common departure of a sequence where the tertiary term is ii and the twelfth term is $\mathrm{-25}.$ Give the formula for the general term.

Find the Sum of the Beginning north Terms of an Arithmetic Sequence

As with the full general sequences, it is oftentimes useful to find the sum of an arithmetic sequence. The sum, ${\mathrm{South}}_n,$ of the offset $n$ terms of whatsoever arithmetic sequence is written as $S_n=a_{\mathrm{ane}}+a_2+a_3+\mathrm{...}+a_n.$ To find the sum by merely adding all the terms tin be dull. So we can also develop a formula to find the sum of a sequence using the first and concluding term of the sequence.

Nosotros can develop this new formula past kickoff writing the sum by starting with the first term, $a_{\mathrm{ane}},$ and keep adding a d to get the next term equally:

$${\mathrm{Southward}}_n=a_1+(a_{\mathrm{i}}+d)+(a_1+\mathrm{two}d)+\dots +a_n.$$

We can also contrary the order of the terms and write the sum by starting with $a_{\mathrm{north}}$ and keep subtracting d to get the next term every bit

$${\mathrm{Due\; south}}_n=a_n+(a_{\mathrm{northward}}–d)+(a_{\mathrm{due\; north}}–2d)+\dots +a_1.$$

If we add together these ii expressions for the sum of the start north terms of an arithmetics sequence, we tin can derive a formula for the sum of the first n terms of whatsoever arithmetic serial.

$$\begin{array}{c}\underset{\text{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}}{\begin{array}{ccccccccccc}\hfill S_n& =\hfill & a_1\hfill & +\hfill & (a_{\mathrm{one}}+d)\hfill & +\hfill & (a_{\mathrm{one}}+2d)\hfill & +\hfill & \dots \hfill & +\hfill & a_{\mathrm{northward}}\hfill \\ +{\mathrm{Due\; south}}_n\hfill & =\hfill & a_n\hfill & +\hfill & (a_n-d)\hfill & +\hfill & (a_{\mathrm{due\; north}}-2d)\hfill & +\hfill & \dots \hfill & +\hfill & a_{\mathrm{one}}\hfill \end{array}}\hfill \\ \\ 2S_n=\left(a_{\mathrm{i}}+a_n\right)+\left(a_1+a_n\right)+\left(a_{\mathrm{one}}+a_{\mathrm{northward}}\right)+\dots +\left(a_1+a_{\mathrm{north}}\right)\hfill \end{array}$$

Because there are due north sums of $\left(a_1+a_{\mathrm{northward}}\right)$ on the correct side of the equation, we rewrite the right side as $n\left(a_1+a_n\right).$

$$2S_n=\mathrm{northward}(a_{\mathrm{i}}+a_n)$$

Nosotros split up by two to solve for ${\mathrm{South}}_{\mathrm{due\; north}}.$

$${\mathrm{South}}_{\mathrm{north}}=\frac{\mathrm{north}}{\mathrm{two}}(a_{\mathrm{one}}+a_n)$$

This gives us a general formula for the sum of the outset north terms of an arithmetics sequence.

Sum of the Get-go n Terms of an Arithmetic Sequence

The sum, ${\mathrm{Due\; south}}_{\mathrm{north}},$ of the first n terms of an arithmetics sequence is

$$S_n=\frac{\mathrm{due\; north}}{2}(a_1+a_n)$$

where $a_{\mathrm{i}}$ is the first term and $a_{\mathrm{northward}}$ is the nth term.

We employ this formula in the next example where the kickoff few terms of the sequence are given.

Example 12.18

Find the sum of the outset xxx terms of the arithmetic sequence: 8, 13, 18, 23, 28, …

Endeavour It 12.35

Find the sum of the commencement xxx terms of the arithmetic sequence: five, ix, xiii, 17, 21, …

Try It 12.36

Detect the sum of the first thirty terms of the arithmetic sequence: 7, 10, thirteen, 16, 19, …

In the next case, nosotros are given the general term for the sequence and are asked to notice the sum of the first 50 terms.

Example 12.19

Find the sum of the commencement 50 terms of the arithmetics sequence whose general term is $a_n=3n-\mathrm{iv}.$

Try Information technology 12.37

Find the sum of the starting time 50 terms of the arithmetic sequence whose general term is $a_{\mathrm{due\; north}}=2n-5.$

Endeavor It 12.38

Detect the sum of the first 50 terms of the arithmetic sequence whose general term is $a_{\mathrm{due\; north}}=4n+3.$

In the next example we are given the sum in summation notation. To add all the terms would exist tedious, and so we extract the information needed to utilize the formula to find the sum of the first northward terms.

Example 12.20

Find the sum: ${\displaystyle \sum _{i=1}^{25}(4i+\mathrm{vii})}.$

Try Information technology 12.39

Observe the sum: ${\displaystyle \sum _{i=1}^{\mathrm{thirty}}(\mathrm{half\; dozen}i-4}).$

Endeavor It 12.forty

Find the sum: ${\displaystyle \sum _{i=1}^{35}(5i-3}).$

Section 12.2 Exercises

Practise Makes Perfect

Determine if a Sequence is Arithmetic

In the following exercises, determine if each sequence is arithmetics, and if so, indicate the mutual deviation.

77.

$\mathrm{four},12,20,28,36,44\text{,}\text{…}$

78 .

$\mathrm{-seven},\mathrm{-2},\mathrm{iii},8,13,\mathrm{eighteen}\text{,}\text{…}$

79.

$\mathrm{-15},\mathrm{-sixteen},\mathrm{three},12,21,\mathrm{xxx}\text{,}\text{…}$

eighty .

$\mathrm{xi},\mathrm{v},\mathrm{-one},\mathrm{-vii}-\mathrm{thirteen},\mathrm{-nineteen}\text{,}\text{…}$

81.

$\mathrm{viii},5,2,\mathrm{-1},\mathrm{-4},\mathrm{-7}\text{,}\text{…}$

82 .

$15,\mathrm{five},\mathrm{-5},\mathrm{-xv},\mathrm{-25},\mathrm{-35}\text{,}\text{…}$

In the following exercises, write the beginning five terms of each sequence with the given offset term and common difference.

83.

$a_1=11$ and $d=\mathrm{vii}$

84 .

$a_{\mathrm{ane}}=18$ and $d=9$

85.

$a_1=\mathrm{-7}$ and $d=4$

86 .

$a_1=\mathrm{-8}$ and $d=5$

87.

$a_{\mathrm{i}}=14$ and $d=\mathrm{-9}$

88 .

$a_1=\mathrm{-3}$ and $d=\mathrm{-iii}$

Find the General Term (nth Term) of an Arithmetics Sequence

In the following exercises, detect the term described using the data provided.

89.

Find the twenty-start term of a sequence where the beginning term is three and the common difference is eight.

90 .

Find the xx-3rd term of a sequence where the beginning term is six and the common difference is four.

91.

Discover the thirtieth term of a sequence where the first term is $\mathrm{-14}$ and the common departure is v.

92 .

Find the fortieth term of a sequence where the first term is $\mathrm{-19}$ and the common divergence is vii.

93.

Find the sixteenth term of a sequence where the get-go term is 11 and the common departure is $\mathrm{-6}.$

94 .

Detect the fourteenth term of a sequence where the offset term is 8 and the common deviation is $\mathrm{-iii}.$

95.

Find the twentieth term of a sequence where the fifth term is $\mathrm{-4}$ and the common difference is $\mathrm{-2}.$ Requite the formula for the full general term.

96 .

Find the thirteenth term of a sequence where the sixth term is $\mathrm{-1}$ and the mutual deviation is $\mathrm{-4}.$ Give the formula for the full general term.

97.

Find the eleventh term of a sequence where the tertiary term is 19 and the common difference is five. Give the formula for the general term.

98 .

Notice the fifteenth term of a sequence where the tenth term is 17 and the common difference is seven. Give the formula for the full general term.

99.

Find the eighth term of a sequence where the seventh term is $\mathrm{-8}$ and the mutual difference is $\mathrm{-five}.$ Requite the formula for the general term.

100 .

Observe the fifteenth term of a sequence where the tenth term is $\mathrm{-eleven}$ and the common difference is $\mathrm{-iii}.$ Give the formula for the general term.

In the following exercises, find the first term and common difference of the sequence with the given terms. Give the formula for the full general term.

101.

The 2nd term is 14 and the thirteenth term is 47.

102 .

The third term is xviii and the fourteenth term is 73.

103.

The second term is 13 and the 10th term is $\mathrm{-51}.$

104 .

The third term is iv and the tenth term is $\mathrm{-38}$ .

105.

The fourth term is $\mathrm{-vi}$ and the fifteenth term is 27.

106 .

The third term is $\mathrm{-13}$ and the seventeenth term is 15.

Observe the Sum of the Commencement n Terms of an Arithmetics Sequence

In the following exercises, find the sum of the first thirty terms of each arithmetic sequence.

107.

$11,14,17,\mathrm{twenty},23\text{,}\text{…}$

108 .

$12,\mathrm{eighteen},24,30,36\text{,}\text{…}$

109.

$8,\mathrm{five},2,\mathrm{-1},\mathrm{-iv}\text{,}\text{…}$

110 .

$16,10,4,\mathrm{-2},\mathrm{-8}\text{,}\text{…}$

111.

$\mathrm{-17},\mathrm{-15},\mathrm{-thirteen},\mathrm{-eleven},\mathrm{-nine}\text{,}\text{…}$

112 .

$\mathrm{-15},\mathrm{-12},\mathrm{-ix},\mathrm{-half\; dozen},\mathrm{-three}\text{,}\text{…}$

In the following exercises, find the sum of the outset 50 terms of the arithmetic sequence whose general term is given.

113.

$a_{\mathrm{north}}=\mathrm{v}n-1$

114 .

$a_n=2\mathrm{due\; north}+7$

115.

$a_n=\mathrm{-3}n+5$

116 .

$a_{\mathrm{north}}=\mathrm{-4}n+\mathrm{iii}$

In the post-obit exercises, find each sum.

117.

${\displaystyle \sum _{i=\mathrm{ane}}^{40}(8i-7})$

118 .

${\displaystyle \sum _{i=1}^{45}(7i-5})$

119.

${\displaystyle \sum _{i=1}^{50}(3i+\mathrm{half-dozen}})$

120 .

${\displaystyle \sum _{i=\mathrm{one}}^{25}(\mathrm{iv}i+3})$

121.

${\displaystyle \sum _{i=1}^{35}(\mathrm{-six}i}-2)$

122 .

${\displaystyle \sum _{i=\mathrm{i}}^{30}(\mathrm{-5}i+\mathrm{i}})$

Writing Exercises

123.

In your own words, explain how to make up one's mind whether a sequence is arithmetic.

124 .

In your ain words, explicate how the commencement ii terms are used to find the tenth term. Testify an example to illustrate your explanation.

125.

In your own words, explain how to find the general term of an arithmetic sequence.

126 .

In your own words, explain how to find the sum of the kickoff $\mathrm{due\; north}$ terms of an arithmetic sequence without calculation all the terms.

Self Check

ⓐ Subsequently completing the exercises, utilise this checklist to evaluate your mastery of the objectives of this section.

ⓑ After reviewing this checklist, what will you practise to become confident for all objectives?

12.1 Arithmetic Sequences Answer Key,

Source: https://openstax.org/books/intermediate-algebra-2e/pages/12-2-arithmetic-sequences

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