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Study Guides > Precalculus II

Arithmetic Sequences

The values of the truck in the example are said to form an arithmetic sequence because they change by a constant amount each year. Each term increases or decreases by the same constant value called the common difference of the sequence. For this sequence, the common difference is –3,400. A sequence, {25000, 21600, 18200, 14800, 8000}, that shows the terms differ only by -3400. The sequence below is another example of an arithmetic sequence. In this case, the constant difference is 3. You can choose any term of the sequence, and add 3 to find the subsequent term. A sequence {3, 6, 9, 12, 15, ...} that shows the terms only differ by 3.

A General Note: Arithmetic Sequence

An arithmetic sequence is a sequence that has the property that the difference between any two consecutive terms is a constant. This constant is called the common difference. If a1{a}_{1} is the first term of an arithmetic sequence and dd is the common difference, the sequence will be:
{an}={a1,a1+d,a1+2d,a1+3d,...}\left\{{a}_{n}\right\}=\left\{{a}_{1},{a}_{1}+d,{a}_{1}+2d,{a}_{1}+3d,...\right\}

Example 1: Finding Common Differences

Is each sequence arithmetic? If so, find the common difference.
  1. {1,2,4,8,16,...}\left\{1,2,4,8,16,...\right\}
  2. {3,1,5,9,13,...}\left\{-3,1,5,9,13,...\right\}

Solution

Subtract each term from the subsequent term to determine whether a common difference exists.
  1. The sequence is not arithmetic because there is no common difference.2 minus 1 = 1. 4 minus 2 = 2. 8 minus 4 = 4. 16 minus 8 equals 8.
  2. The sequence is arithmetic because there is a common difference. The common difference is 4.1 minus negative 3 equals 4. 5 minus 1 equals 4. 9 minus 5 equals 4. 13 minus 9 equals 4.

Analysis of the Solution

The graph of each of these sequences is shown in Figure 1. We can see from the graphs that, although both sequences show growth, aa is not linear whereas bb is linear. Arithmetic sequences have a constant rate of change so their graphs will always be points on a line.
Two graphs of arithmetic sequences. Graph (a) grows exponentially while graph (b) grows linearly. Figure 1

Q & A

If we are told that a sequence is arithmetic, do we have to subtract every term from the following term to find the common difference?

No. If we know that the sequence is arithmetic, we can choose any one term in the sequence, and subtract it from the subsequent term to find the common difference.

Try It 1

Is the given sequence arithmetic? If so, find the common difference.
{18, 16, 14, 12, 10,}\left\{18,\text{ }16,\text{ }14,\text{ }12,\text{ }10,\dots \right\}
Solution

Try It 2

Is the given sequence arithmetic? If so, find the common difference.
{1, 3, 6, 10, 15,}\left\{1,\text{ }3,\text{ }6,\text{ }10,\text{ }15,\dots \right\}
Solution

Writing Terms of Arithmetic Sequences

Now that we can recognize an arithmetic sequence, we will find the terms if we are given the first term and the common difference. The terms can be found by beginning with the first term and adding the common difference repeatedly. In addition, any term can also be found by plugging in the values of nn and dd into formula below.
an=a1+(n1)d{a}_{n}={a}_{1}+\left(n - 1\right)d

How To: Given the first term and the common difference of an arithmetic sequence, find the first several terms.

  1. Add the common difference to the first term to find the second term.
  2. Add the common difference to the second term to find the third term.
  3. Continue until all of the desired terms are identified.
  4. Write the terms separated by commas within brackets.

Example 2: Writing Terms of Arithmetic Sequences

Write the first five terms of the arithmetic sequence with a1=17{a}_{1}=17 and d=3d=-3 .

Solution

Adding 3-3 is the same as subtracting 3. Beginning with the first term, subtract 3 from each term to find the next term. The first five terms are {17,14,11,8,5}\left\{17,14,11,8,5\right\}

Analysis of the Solution

As expected, the graph of the sequence consists of points on a line as shown in Figure 2.
Graph of the arithmetic sequence. The points form a negative line. Figure 2

Try It 3

List the first five terms of the arithmetic sequence with a1=1{a}_{1}=1 and d=5d=5 . Solution

How To: Given any the first term and any other term in an arithmetic sequence, find a given term.

  1. Substitute the values given for a1,an,n{a}_{1},{a}_{n},n into the formula an=a1+(n1)d{a}_{n}={a}_{1}+\left(n - 1\right)d to solve for dd.
  2. Find a given term by substituting the appropriate values for a1,n{a}_{1},n, and dd into the formula an=a1+(n1)d{a}_{n}={a}_{1}+\left(n - 1\right)d.

Example 3: Writing Terms of Arithmetic Sequences

Given a1=8{a}_{1}=8 and a4=14{a}_{4}=14 , find a5{a}_{5} .

Solution

The sequence can be written in terms of the initial term 8 and the common difference dd .
{8,8+d,8+2d,8+3d}\left\{8,8+d,8+2d,8+3d\right\}
We know the fourth term equals 14; we know the fourth term has the form a1+3d=8+3d{a}_{1}+3d=8+3d . We can find the common difference dd .
an=a1+(n1)da4=a1+3da4=8+3dWrite the fourth term of the sequence in terms of a1 and d.14=8+3dSubstitute 14 for a4.d=2Solve for the common difference.\begin{array}{ll}{a}_{n}={a}_{1}+\left(n - 1\right)d\hfill & \hfill \\ {a}_{4}={a}_{1}+3d\hfill & \hfill \\ {a}_{4}=8+3d\hfill & \text{Write the fourth term of the sequence in terms of } {a}_{1} \text{ and } d.\hfill \\ 14=8+3d\hfill & \text{Substitute } 14 \text{ for } {a}_{4}.\hfill \\ d=2\hfill & \text{Solve for the common difference}.\hfill \end{array}
Find the fifth term by adding the common difference to the fourth term.
a5=a4+2=16{a}_{5}={a}_{4}+2=16

Analysis of the Solution

Notice that the common difference is added to the first term once to find the second term, twice to find the third term, three times to find the fourth term, and so on. The tenth term could be found by adding the common difference to the first term nine times or by using the equation an=a1+(n1)d{a}_{n}={a}_{1}+\left(n - 1\right)d.

Try It 4

Given a3=7{a}_{3}=7 and a5=17{a}_{5}=17 , find a2{a}_{2} . Solution

Using Formulas for Arithmetic Sequences

Some arithmetic sequences are defined in terms of the previous term using a recursive formula. The formula provides an algebraic rule for determining the terms of the sequence. A recursive formula allows us to find any term of an arithmetic sequence using a function of the preceding term. Each term is the sum of the previous term and the common difference. For example, if the common difference is 5, then each term is the previous term plus 5. As with any recursive formula, the first term must be given.
an=an1+dn2\begin{array}{lllll}{a}_{n}={a}_{n - 1}+d\hfill & \hfill & \hfill & \hfill & n\ge 2\hfill \end{array}

A General Note: Recursive Formula for an Arithmetic Sequence

The recursive formula for an arithmetic sequence with common difference dd is:
an=an1+dn2\begin{array}{lllll}{a}_{n}={a}_{n - 1}+d\hfill & \hfill & \hfill & \hfill & n\ge 2\hfill \end{array}

How To: Given an arithmetic sequence, write its recursive formula.

  1. Subtract any term from the subsequent term to find the common difference.
  2. State the initial term and substitute the common difference into the recursive formula for arithmetic sequences.

Example 4: Writing a Recursive Formula for an Arithmetic Sequence

Write a recursive formula for the arithmetic sequence.
{18741526}\left\{-18\text{, }-7\text{, }4\text{, }15\text{, }26\text{, \ldots }\right\}

Solution

The first term is given as 18-18 . The common difference can be found by subtracting the first term from the second term.
d=7(18)=11d=-7-\left(-18\right)=11
Substitute the initial term and the common difference into the recursive formula for arithmetic sequences.
a1=18an=an1+11, for n2\begin{array}{l}{a}_{1}=-18\hfill \\ {a}_{n}={a}_{n - 1}+11,\text{ for }n\ge 2\hfill \end{array}

Analysis of the Solution

We see that the common difference is the slope of the line formed when we graph the terms of the sequence, as shown in Figure 3. The growth pattern of the sequence shows the constant difference of 11 units.
Graph of the arithmetic sequence. The points form a positive line. Figure 3

How To: Do we have to subtract the first term from the second term to find the common difference?

No. We can subtract any term in the sequence from the subsequent term. It is, however, most common to subtract the first term from the second term because it is often the easiest method of finding the common difference.

Try It 5

Write a recursive formula for the arithmetic sequence.
{25374961}\left\{25\text{, } 37\text{, } 49\text{, } 61\text{, } \text{\ldots }\right\}
Solution

Using Explicit Formulas for Arithmetic Sequences

We can think of an arithmetic sequence as a function on the domain of the natural numbers; it is a linear function because it has a constant rate of change. The common difference is the constant rate of change, or the slope of the function. We can construct the linear function if we know the slope and the vertical intercept.
an=a1+d(n1){a}_{n}={a}_{1}+d\left(n - 1\right)
To find the y-intercept of the function, we can subtract the common difference from the first term of the sequence. Consider the following sequence. A sequence, {200, 150, 100, 50, 0, ...}, that shows the terms differ only by -50. The common difference is 50-50 , so the sequence represents a linear function with a slope of 50-50 . To find the yy -intercept, we subtract 50-50 from 200:200(50)=200+50=250200:200-\left(-50\right)=200+50=250 . You can also find the yy -intercept by graphing the function and determining where a line that connects the points would intersect the vertical axis. The graph is shown in Figure 4.
Graph of the arithmetic sequence. The points form a negative line. Figure 4
Recall the slope-intercept form of a line is y=mx+by=mx+b. When dealing with sequences, we use an{a}_{n} in place of yy and nn in place of xx. If we know the slope and vertical intercept of the function, we can substitute them for mm and bb in the slope-intercept form of a line. Substituting 50-50 for the slope and 250250 for the vertical intercept, we get the following equation:
an=50n+250{a}_{n}=-50n+250
We do not need to find the vertical intercept to write an explicit formula for an arithmetic sequence. Another explicit formula for this sequence is an=20050(n1){a}_{n}=200 - 50\left(n - 1\right) , which simplifies to an=50n+250{a}_{n}=-50n+250.

A General Note: Explicit Formula for an Arithmetic Sequence

An explicit formula for the nthn\text{th} term of an arithmetic sequence is given by
an=a1+d(n1){a}_{n}={a}_{1}+d\left(n - 1\right)

How To: Given the first several terms for an arithmetic sequence, write an explicit formula.

  1. Find the common difference, a2a1{a}_{2}-{a}_{1}.
  2. Substitute the common difference and the first term into an=a1+d(n1){a}_{n}={a}_{1}+d\left(n - 1\right).

Example 5: Writing the nth Term Explicit Formula for an Arithmetic Sequence

Write an explicit formula for the arithmetic sequence.
{212223242}\left\{2\text{, }12\text{, }22\text{, }32\text{, }42\text{, \ldots }\right\}

Solution

The common difference can be found by subtracting the first term from the second term.
d=a2a1=122=10\begin{array}{ll}d\hfill & ={a}_{2}-{a}_{1}\hfill \\ \hfill & =12 - 2\hfill \\ \hfill & =10\hfill \end{array}
The common difference is 10. Substitute the common difference and the first term of the sequence into the formula and simplify.
an=2+10(n1)an=10n8\begin{array}{l}{a}_{n}=2+10\left(n - 1\right)\hfill \\ {a}_{n}=10n - 8\hfill \end{array}

Analysis of the Solution

The graph of this sequence, represented in Figure 5, shows a slope of 10 and a vertical intercept of 8-8 .
Graph of the arithmetic sequence. The points form a positive line. Figure 5

Try It 6

Write an explicit formula for the following arithmetic sequence.
{50,47,44,41,}\left\{50,47,44,41,\dots \right\}
Solution

Finding the Number of Terms in a Finite Arithmetic Sequence

Explicit formulas can be used to determine the number of terms in a finite arithmetic sequence. We need to find the common difference, and then determine how many times the common difference must be added to the first term to obtain the final term of the sequence.

How To: Given the first three terms and the last term of a finite arithmetic sequence, find the total number of terms.

  1. Find the common difference dd.
  2. Substitute the common difference and the first term into an=a1+d(n1){a}_{n}={a}_{1}+d\left(n - 1\right).
  3. Substitute the last term for an{a}_{n} and solve for nn.

Example 6: Finding the Number of Terms in a Finite Arithmetic Sequence

Find the number of terms in the finite arithmetic sequence.
{816...41}\left\{8\text{, }1\text{, }-6\text{, }...\text{, }-41\right\}

Solution

The common difference can be found by subtracting the first term from the second term.
18=71 - 8=-7
The common difference is 7-7 . Substitute the common difference and the initial term of the sequence into the nthn\text{th} term formula and simplify.
an=a1+d(n1)an=8+7(n1)an=157n\begin{array}{l}{a}_{n}={a}_{1}+d\left(n - 1\right)\hfill \\ {a}_{n}=8+-7\left(n - 1\right)\hfill \\ {a}_{n}=15 - 7n\hfill \end{array}
Substitute 41-41 for an{a}_{n} and solve for nn
41=157n8=n\begin{array}{l}-41=15 - 7n\hfill \\ 8=n\hfill \end{array}
There are eight terms in the sequence.

Try It 7

Find the number of terms in the finite arithmetic sequence.
{61116...56}\left\{6\text{, }11\text{, }16\text{, }...\text{, }56\right\}
Solution

Solving Application Problems with Arithmetic Sequences

In many application problems, it often makes sense to use an initial term of a0{a}_{0} instead of a1{a}_{1}. In these problems, we alter the explicit formula slightly to account for the difference in initial terms. We use the following formula:
an=a0+dn{a}_{n}={a}_{0}+dn

Example 7: 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?

Solution

  1. The situation can be modeled by an arithmetic sequence with an initial term of 1 and a common difference of 2.Let AA be the amount of the allowance and nn be the number of years after age 5. Using the altered explicit formula for an arithmetic sequence we get:
    An=1+2n{A}_{n}=1+2n
  2. We can find the number of years since age 5 by subtracting.
    165=1116 - 5=11
    We are looking for the child’s allowance after 11 years. Substitute 11 into the formula to find the child’s allowance at age 16.
    A11=1+2(11)=23{A}_{11}=1+2\left(11\right)=23
    The child’s allowance at age 16 will be $23 per week.

Try It 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

Key Equations

recursive formula for nth term of an arithmetic sequence {a}_{n}={a}_{n - 1}+d\phantom{\rule{1}{0ex}}n\ge 2
explicit formula for nth term of an arithmetic sequence an=a1+d(n1)\begin{array}{l}{a}_{n}={a}_{1}+d\left(n - 1\right)\end{array}

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.
  • The terms of an arithmetic sequence can be found by beginning with the initial term and adding the common difference repeatedly.
  • A recursive formula for an arithmetic sequence with common difference dd is given by an=an1+d,n2{a}_{n}={a}_{n - 1}+d,n\ge 2.
  • As with any recursive formula, the initial term of the sequence must be given.
  • An explicit formula for an arithmetic sequence with common difference dd is given by an=a1+d(n1){a}_{n}={a}_{1}+d\left(n - 1\right).
  • An explicit formula can be used to find the number of terms in a sequence.
  • In application problems, we sometimes alter the explicit formula slightly to an=a0+dn{a}_{n}={a}_{0}+dn.

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

Section Exercises

1. What is an arithmetic sequence? 2. How is the common difference of an arithmetic sequence found? 3. How do we determine whether a sequence is arithmetic? 4. What are the main differences between using a recursive formula and using an explicit formula to describe an arithmetic sequence? 5. Describe how linear functions and arithmetic sequences are similar. How are they different? For the following exercises, find the common difference for the arithmetic sequence provided. 6. {5,11,17,23,29,...}\left\{5,11,17,23,29,...\right\} 7. {0,12,1,32,2,...}\left\{0,\frac{1}{2},1,\frac{3}{2},2,...\right\} For the following exercises, determine whether the sequence is arithmetic. If so find the common difference. 8. {11.4,9.3,7.2,5.1,3,...}\left\{11.4,9.3,7.2,5.1,3,...\right\} 9. {4,16,64,256,1024,...}\left\{4,16,64,256,1024,...\right\} For the following exercises, write the first five terms of the arithmetic sequence given the first term and common difference. 10. a1=25{a}_{1}=-25 , d=9d=-9 11. a1=0{a}_{1}=0 , d=23d=\frac{2}{3} For the following exercises, write the first five terms of the arithmetic series given two terms. 12. a1=17,a7=31{a}_{1}=17,{a}_{7}=-31 13. a13=60,a33=160{a}_{13}=-60,{a}_{33}=-160 For the following exercises, find the specified term for the arithmetic sequence given the first term and common difference. 14. First term is 3, common difference is 4, find the 5th term. 15. First term is 4, common difference is 5, find the 4th term. 16. First term is 5, common difference is 6, find the 8th term. 17. First term is 6, common difference is 7, find the 6th term. 18. 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. 19. Find the first term or a1{a}_{1} of an arithmetic sequence if a6=12{a}_{6}=12 and a14=28{a}_{14}=28. 20. Find the first term or a1{a}_{1} of an arithmetic sequence if a7=21{a}_{7}=21 and a15=42{a}_{15}=42. 21. Find the first term or a1{a}_{1} of an arithmetic sequence if a8=40{a}_{8}=40 and a23=115{a}_{23}=115. 22. Find the first term or a1{a}_{1} of an arithmetic sequence if a9=54{a}_{9}=54 and a17=102{a}_{17}=102. 23. Find the first term or a1{a}_{1} of an arithmetic sequence if a11=11{a}_{11}=11 and a21=16{a}_{21}=16. For the following exercises, find the specified term given two terms from an arithmetic sequence. 24. a1=33{a}_{1}=33 and a7=15{a}_{7}=-15. Find a4{a}_{4}. 25. a3=17.1{a}_{3}=-17.1 and a10=15.7{a}_{10}=-15.7. Find a21{a}_{21}. For the following exercises, use the recursive formula to write the first five terms of the arithmetic sequence. 26. a1=39; an=an13{a}_{1}=39;\text{ }{a}_{n}={a}_{n - 1}-3 27. a1=19; an=an11.4{a}_{1}=-19;\text{ }{a}_{n}={a}_{n - 1}-1.4 For the following exercises, write a recursive formula for each arithmetic sequence. 28. an={40,60,80,...}{a}_{n}=\left\{40,60,80,...\right\} 29. an={17,26,35,...}{a}_{n}=\left\{17,26,35,...\right\} 30. an={1,2,5,...}{a}_{n}=\left\{-1,2,5,...\right\} 31. an={12,17,22,...}{a}_{n}=\left\{12,17,22,...\right\} 32. an={15,7,1,...}{a}_{n}=\left\{-15,-7,1,...\right\} 33. an={8.9,10.3,11.7,...}{a}_{n}=\left\{8.9,10.3,11.7,...\right\} 34. an={0.52,1.02,1.52,...}{a}_{n}=\left\{-0.52,-1.02,-1.52,...\right\} 35. an={15,920,710,...}{a}_{n}=\left\{\frac{1}{5},\frac{9}{20},\frac{7}{10},...\right\} 36. an={12,54,2,...}{a}_{n}=\left\{-\frac{1}{2},-\frac{5}{4},-2,...\right\} 37. an={16,1112,2,...}{a}_{n}=\left\{\frac{1}{6},-\frac{11}{12},-2,...\right\} For the following exercises, write a recursive formula for the given arithmetic sequence, and then find the specified term. 38. an={741...}{a}_{n}=\left\{7\text{, }4\text{, }1\text{, }...\right\}; Find the 17th term. 39. an={41118...}{a}_{n}=\left\{4\text{, }11\text{, }18\text{, }...\right\}; Find the 14th term. 40. an={2610...}{a}_{n}=\left\{2\text{, }6\text{, }10\text{, }...\right\}; Find the 12th term. For the following exercises, use the explicit formula to write the first five terms of the arithmetic sequence. 41. an=244n{a}_{n}=24 - 4n 42. an=12n12{a}_{n}=\frac{1}{2}n-\frac{1}{2} For the following exercises, write an explicit formula for each arithmetic sequence. 43. an={3,5,7,...}{a}_{n}=\left\{3,5,7,...\right\} 44. an={32,24,16,...}{a}_{n}=\left\{32,24,16,...\right\} 45. an={595195...}{a}_{n}=\left\{-5\text{, }95\text{, }195\text{, }...\right\} 46. an={17217417,...}{a}_{n}=\left\{-17\text{, }-217\text{, }-417\text{,}...\right\} 47. an={1.83.65.4...}{a}_{n}=\left\{1.8\text{, }3.6\text{, }5.4\text{, }...\right\} 48. an={18.1,16.2,14.3,...}{a}_{n}=\left\{-18.1,-16.2,-14.3,...\right\} 49. an={15.8,18.5,21.2,...}{a}_{n}=\left\{15.8,18.5,21.2,...\right\} 50. an={13,43,3...}{a}_{n}=\left\{\frac{1}{3},-\frac{4}{3},-3\text{, }...\right\} 51. an={0,13,23,...}{a}_{n}=\left\{0,\frac{1}{3},\frac{2}{3},...\right\} 52. an={5,103,53,}{a}_{n}=\left\{-5,-\frac{10}{3},-\frac{5}{3},\dots \right\} For the following exercises, find the number of terms in the given finite arithmetic sequence. 53. an={3,4,11...,60}{a}_{n}=\left\{3\text{,}-4\text{,}-11\text{, }...\text{,}-60\right\} 54. an={1.2,1.4,1.6,...,3.8}{a}_{n}=\left\{1.2,1.4,1.6,...,3.8\right\} 55. an={12,2,72,...,8}{a}_{n}=\left\{\frac{1}{2},2,\frac{7}{2},...,8\right\} For the following exercises, determine whether the graph shown represents an arithmetic sequence. 56. Graph of a scattered plot with labeled points: (1, -4), (2, -2), (3, 0), (4, 2), and (5, 4). The x-axis is labeled n and the y-axis is labeled a_n. 57. Graph of a scattered plot with labeled points: (1, 1.5), (2, 2.25), (3, 3.375), (4, 5.0625), and (5, 7.5938). The x-axis is labeled n and the y-axis is labeled a_n. For the following exercises, use the information provided to graph the first 5 terms of the arithmetic sequence. 58. a1=0,d=4{a}_{1}=0,d=4 59. a1=9;an=an110{a}_{1}=9;{a}_{n}={a}_{n - 1}-10 60. an=12+5n{a}_{n}=-12+5n For the following exercises, follow the steps to work with the arithmetic sequence an=3n2{a}_{n}=3n - 2 using a graphing calculator:
  • Press [MODE]
    • Select SEQ in the fourth line
    • Select DOT in the fifth line
    • Press [ENTER]
  • Press [Y=]
    • nMinn\text{Min} is the first counting number for the sequence. Set nMin=1n\text{Min}=1
    • u(n)u\left(n\right) is the pattern for the sequence. Set u(n)=3n2u\left(n\right)=3n - 2
    • u(nMin)u\left(n\text{Min}\right) is the first number in the sequence. Set u(nMin)=1u\left(n\text{Min}\right)=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
61. What are the first seven terms shown in the column with the heading u(n)?u\left(n\right)\text{?} 62. Use the scroll-down arrow to scroll to n=50n=50. What value is given for u(n)?u\left(n\right)\text{?} 63. Press [WINDOW]. Set nMin=1,nMax=5,xMin=0,xMax=6,yMin=1n\text{Min}=1,n\text{Max}=5,x\text{Min}=0,x\text{Max}=6,y\text{Min}=-1, and yMax=14y\text{Max}=14. Then press [GRAPH]. Graph the sequence as it appears on the graphing calculator. For the following exercises, follow the steps given above to work with the arithmetic sequence an=12n+5{a}_{n}=\frac{1}{2}n+5 using a graphing calculator. 64. What are the first seven terms shown in the column with the heading u(n)u\left(n\right) in the TABLE feature? 65. Graph the sequence as it appears on the graphing calculator. Be sure to adjust the WINDOW settings as needed. 66. Give two examples of arithmetic sequences whose 4th terms are 99. 67. Give two examples of arithmetic sequences whose 10th terms are 206206. 68. Find the 5th term of the arithmetic sequence {9b,5b,b,}\left\{9b,5b,b,\dots \right\}. 69. Find the 11th term of the arithmetic sequence {3a2b,a+2b,a+6b}\left\{3a - 2b,a+2b,-a+6b\dots \right\}. 70. At which term does the sequence {5.4,14.5,23.6,...}\left\{5.4,14.5,23.6,...\right\} exceed 151? 71. At which term does the sequence {173,316,143,...}\left\{\frac{17}{3},\frac{31}{6},\frac{14}{3},...\right\} begin to have negative values? 72. For which terms does the finite arithmetic sequence {52,198,94,...,18}\left\{\frac{5}{2},\frac{19}{8},\frac{9}{4},...,\frac{1}{8}\right\} have integer values? 73. Write an arithmetic sequence using a recursive formula. Show the first 4 terms, and then find the 31st term. 74. Write an arithmetic sequence using an explicit formula. Show the first 4 terms, and then find the 28th term.

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