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Section 10.2 Arithmetic Sequences
10.2 Arithmetic Sequences
Objectives
Y
our grandmother and her financial counselor
are looking at options in case an adult residential facility is needed in the future. The
good news is that your grandmother’s
total assets are $500,000. The bad
news is that adult residential community costs average $64,130
annually, increasing by $1800
each year. In this section, we
will see how sequences can
be used to model your
grandmother’s situation
and help her to identify
realistic options.
� Find the common difference
�
�
�
�
for an arithmetic sequence.
Write terms of an arithmetic
sequence.
Use the formula for the
general term of an arithmetic
sequence.
Use the formula for the sum
of the first n terms of an
arithmetic sequence.
Find the common difference for
an arithmetic sequence.
Arithmetic Sequences
The bar graph in Figure 10.2 shows
annual salaries, rounded to the
nearest thousand dollars, of U.S.
senators from 2000 to 2005. The
graph illustrates that each year
salaries increased by $4 thousand.
The sequence of annual salaries
142, 146, 150, 154, 158, 162, Á
Annual Salaries of U.S. Senators
162
162
Annual Salary
(thousands of dollars)
Section
963
158
158
154
154
150
150
146
146
142
142
shows that each term after the first,
142, differs from the preceding
term by a constant amount, namely
2000 2001 2002 2003
Year
4. This sequence is an example of
Figure 10.2 Source: U.S. Senate
an arithmetic sequence.
2004
2005
Definition of an Arithmetic Sequence
An arithmetic sequence is a sequence in which each term after the first differs from
the preceding term by a constant amount. The difference between consecutive
terms is called the common difference of the sequence.
The common difference, d, is found by subtracting any term from the term
that directly follows it. In the following examples, the common difference is found
by subtracting the first term from the second term, a 2 - a1 .
Arithmetic Sequence
Common Difference
142, 146, 150, 154, 158, Á
-5, - 2, 1, 4, 7, Á
8, 3, -2, - 7, -12, Á
d = 146 - 142 = 4
d = - 2 - 1- 52 = - 2 + 5 = 3
d = 3 - 8 = -5
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964 Chapter 10 Sequences, Induction, and Probability
Figure 10.3 shows the graphs of the last two arithmetic sequences in our list.
The common difference for the increasing sequence in Figure 10.3(a) is 3. The
common difference for the decreasing sequence in Figure 10.3(b) is -5.
an
7
5
3
1
−1
−3
−5
−7
−9
−11
−13
bn
Constant
term-to-term
change
is 3.
2 4 6 8 10
First term is 8.
8
6
4
2
n
2 4 6 8 10
−2
−4
−6
−8
−10
−12
First term is −5.
Figure 10.3(a) The graph
of 5a n6 = - 5, -2, 1, 4, 7, Á
n
Constant
term-to-term
change
is −5.
Figure 10.3(b) The graph of
5bn6 = 8, 3, - 2, -7, -12, Á
The graph of each arithmetic sequence in Figure 10.3 forms a set of discrete
points lying on a straight line. This illustrates that an arithmetic sequence is a linear
function whose domain is the set of positive integers.
If the first term of an arithmetic sequence is a 1 , each term after the first is
obtained by adding d, the common difference, to the previous term. This can be
expressed recursively as follows:
an=an – 1+d.
Add d to the term in any
position to get the next term.
To use this recursion formula, we must be given the first term.
�
Write terms of an arithmetic
sequence.
Writing the Terms of an Arithmetic Sequence
EXAMPLE 1
Write the first six terms of the arithmetic sequence in which a1 = 6 and
an = an - 1 - 2.
Solution
The recursion formula a1 = 6 and an = an - 1 - 2 indicates that each
term after the first, 6, is obtained by adding -2 to the previous term.
a1 = 6
a2 = a1 - 2 = 6 - 2
a3 = a2 - 2 = 4 - 2
= 4
= 2
Use an = an - 1 - 2 with n = 2.
a4 = a3 - 2 = 2 - 2
= 0
Use an = an - 1 - 2 with n = 4.
This is given.
a5 = a4 - 2 = 0 - 2 = - 2
a6 = a5 - 2 = - 2 - 2 = - 4
Use an = an - 1 - 2 with n = 3.
Use an = an - 1 - 2 with n = 5.
Use an = an - 1 - 2 with n = 6.
The first six terms are
6, 4, 2, 0, -2, and - 4.
Check Point
1
Write the first six terms of the arithmetic sequence in which
a 1 = 100 and an = an - 1 - 30.
�
Use the formula for the general
term of an arithmetic sequence.
The General Term of an Arithmetic Sequence
Consider an arithmetic sequence whose first term is a1 and whose common difference
is d. We are looking for a formula for the general term, a n . Let’s begin by writing the
first six terms. The first term is a 1 . The second term is a 1 + d. The third term is
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Section 10.2 Arithmetic Sequences
965
a 1 + d + d, or a1 + 2d. Thus, we start with a 1 and add d to each successive term. The
first six terms are
a1,
a1, first
term
a1+d,
a1+2d,
a1+3d,
a1+4d,
a1+5d.
a2, second
term
a3, third
term
a4, fourth
term
a5, fifth
term
a6, sixth
term
Compare the coefficient of d and the subscript of a denoting the term number. Can you
see that the coefficient of d is 1 less than the subscript of a denoting the term number?
a3: third term=a1+2d
a4: fourth term=a1+3d
One less than 3, or 2, is
the coefficient of d.
One less than 4, or 3, is
the coefficient of d.
Thus, the formula for the nth term is
an: nth term=a1+(n-1)d.
One less than n, or n − 1, is
the coefficient of d.
General Term of an Arithmetic Sequence
The nth term (the general term) of an arithmetic sequence with first term a1 and
common difference d is
a n = a1 + 1n - 12d.
Using the Formula for the General Term
of an Arithmetic Sequence
EXAMPLE 2
Find the eighth term of the arithmetic sequence whose first term is 4 and whose
common difference is -7.
Solution To find the eighth term, a8 , we replace n in the formula with 8, a1 with
4, and d with - 7.
an = a1 + 1n - 12d
a8 = 4 + 18 - 121 - 72 = 4 + 71-72 = 4 + 1- 492 = - 45
The eighth term is - 45. We can check this result by writing the first eight terms of
the sequence:
4, -3, - 10, - 17, -24, -31, -38, -45.
Check Point
2
Find the ninth term of the arithmetic sequence whose first term
is 6 and whose common difference is -5.
EXAMPLE 3
Using an Arithmetic Sequence to Model
Teachers’ Earnings
According to the National Education Association, teachers in the United States
earned an average of $44,600 in 2002. This amount has increased by approximately
$1130 per year.
a. Write a formula for the nth term of the arithmetic sequence that describes
teachers’ average earnings n years after 2001.
b. How much will U.S. teachers earn, on average, by the year 2012?
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Solution
a. With a yearly increase of $1130, we can express teachers’ earnings by the
following arithmetic sequence:
44,600,
44,600+1130=45,730,
a1: earnings in 2002,
1 year after 2001
45,730+1130=46,860, » .
a2: earnings in 2003,
2 years after 2001
a3: earnings in 2004,
3 years after 2001
In the sequence 44,600, 45,730, 46,860, Á , a 1 , the first term, represents the
amount teachers earned in 2002. Each subsequent year this amount increases
by $1130, so d = 1130. We use the formula for the general term of an arithmetic sequence to write the nth term of the sequence that describes teachers’
earnings n years after 2001.
an = a1 + 1n - 12d
This is the formula for the general term of an
arithmetic sequence.
an = 44,600 + 1n - 121130
a1 = 44,600 and d = 1130.
an = 44,600 + 1130n - 1130
Distribute 1130 to each term in parentheses.
an = 1130n + 43,470
Simplify.
Thus, teachers’ earnings n years after 2001 can be described by
an = 1130n + 43,470.
b. Now we need to project teachers’ earnings in 2012. The year 2012 is 11 years
after 2001. Thus, n = 11. We substitute 11 for n in an = 1130n + 43,470.
a11 = 1130 # 11 + 43,470 = 55,900
The 11th term of the sequence is 55,900. Thus, U.S. teachers are projected to
earn an average of $55,900 by the year 2012.
3
Check Point Thanks to drive-thrus and curbside delivery, Americans are
eating more meals behind the wheel. In 2004, we averaged 32 à la car meals,
increasing by approximately 0.7 meal per year. (Source: Newsweek)
a. Write a formula for the nth term of the arithmetic sequence that models the
average number of car meals n years after 2003.
b. How many car meals will Americans average by the year 2014?
�
Use the formula for the sum of
the first n terms of an arithmetic
sequence.
The Sum of the First n Terms of an Arithmetic Sequence
The sum of the first n terms of an arithmetic sequence, denoted by Sn , and called the
nth partial sum, can be found without having to add up all the terms. Let
Sn = a1 + a2 + a3 + Á + an
be the sum of the first n terms of an arithmetic sequence. Because d is the common
difference between terms, Sn can be written forward and backward as follows:
Forward: Start with
the first term, a1.
Keep adding d.
Backward: Start with
the last term, an.
Keep subtracting d.
+(a1+d) +(a1+2d) +. . .+an
+(an-d) +(an-2d) +. . .+a1
2Sn=(a1+an) +(a1+an) +(a1 +an) +. . .+(a1 +an).
Sn=a1
Sn=an
Add the two equations.
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Section 10.2 Arithmetic Sequences
967
Because there are n sums of 1a1 + an2 on the right side, we can express this side as
n1a1 + an2. Thus, the last equation can be written as follows:
2Sn = n1a1 + an2
n
Sn = 1a1 + an2.
2
Solve for Sn , dividing both sides by 2.
We have proved the following result:
The Sum of the First n Terms of an Arithmetic Sequence
The sum, Sn , of the first n terms of an arithmetic sequence is given by
n
Sn = 1a1 + an2,
2
in which a1 is the first term and an is the nth term.
n
1a + an2,
2 1
we need to know the first term, a1 , the last term, an , and the number of terms, n. The
following examples illustrate how to use this formula.
To find the sum of the terms of an arithmetic sequence using Sn =
Finding the Sum of n Terms
of an Arithmetic Sequence
EXAMPLE 4
Find the sum of the first 100 terms of the arithmetic sequence: 1, 3, 5, 7, Á .
Solution By finding the sum of the first 100 terms of 1, 3, 5, 7, Á , we are finding
the sum of the first 100 odd numbers. To find the sum of the first 100 terms, S100 , we
replace n in the formula with 100.
n
Sn= (a1+an)
2
100
S100=
(a1+a100)
2
The first term,
a1, is 1.
We must find a100,
the 100th term.
We use the formula for the general term of a sequence to find a100 . The common
difference, d, of 1, 3, 5, 7, Á , is 2.
an = a1 + 1n - 12d
a100 = 1 + 1100 - 12 # 2
= 1 + 99 # 2
= 1 + 198 = 199
This is the formula for the nth term of an arithmetic
sequence. Use it to find the 100th term.
Substitute 100 for n, 2 for d, and
1 (the first term) for a1 .
Perform the subtraction in parentheses.
Multiply 199 # 2 = 1982 and then add.
Now we are ready to find the sum of the 100 terms 1, 3, 5, 7, Á , 199.
Sn =
S100 =
n
1a + an2
2 1
Use the formula for the sum of the first n terms of an
arithmetic sequence. Let n = 100, a1 = 1, and a100 = 199.
100
11 + 1992 = 5012002 = 10,000
2
The sum of the first 100 odd numbers is 10,000. Equivalently, the 100th partial sum
of the sequence 1, 3, 5, 7, Á is 10,000.
Check Point
3, 6, 9, 12, Á .
4
Find the sum of the first 15 terms of the arithmetic sequence:
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968 Chapter 10 Sequences, Induction, and Probability
Using Sn to Evaluate a Summation
EXAMPLE 5
25
Find the following sum: a 15i - 92.
i=1
Technology
Solution
To find
25
25
a 15i - 92
i=1
#
#
#
Á + 15 # 25 - 92
a 15i - 92 = 15 1 - 92 + 15 2 - 92 + 15 3 - 92 +
i=1
= -4
on a graphing utility, enter
冷SUM 冷 冷SEQ 冷 15x
- 9, x, 1, 25, 12.
Then press 冷ENTER 冷.
+ 1
+ 6
+ Á + 116
By evaluating the first three terms and the last term, we see that a1 = - 4; d, the
common difference, is 1 - 1- 42, or 5; and a25 , the last term, is 116.
Sn =
S25 =
n
1a + an2
2 1
Use the formula for the sum of the first n terms of an
arithmetic sequence. Let n = 25, a1 = - 4, and a25 = 116.
25
25
1-4 + 1162 =
11122 = 1400
2
2
Thus,
25
a 15i - 92 = 1400.
i=1
Check Point
EXAMPLE 6
5
30
Find the following sum: a 16i - 112.
i=1
Modeling Total Residential Community Costs
over a Six-Year Period
Your grandmother has assets of $500,000. One option that she is considering involves
an adult residential community for a six-year period beginning in 2009. The model
an = 1800n + 64,130
describes yearly adult residential community costs n years after 2008. Does your
grandmother have enough to pay for the facility?
Solution We must find the sum of an arithmetic sequence whose general term is
an = 1800n + 64,130. The first term of the sequence corresponds to the facility’s
costs in the year 2009. The last term corresponds to costs in the year 2014. Because
the model describes costs n years after 2008, n = 1 describes the year 2009 and
n = 6 describes the year 2014.
an = 1800n + 64,130
This is the given formula for the
general term of the sequence.
a1 = 1800 # 1 + 64,130 = 65,930
Find a1 by replacing n with 1.
a6 = 1800 # 6 + 64,130 = 74,930
Find a6 by replacing n with 6.
The first year the facility will cost $65,930. By year six, the facility will cost $74,930.
Now we must find the sum of the costs for all six years. We focus on the sum of the
first six terms of the arithmetic sequence
65,930, 67,730, . . . , 74,930.
a1
a2
a6
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Section 10.2 Arithmetic Sequences
969
We find this sum using the formula for the sum of the first n terms of an arithmetic
sequence. We are adding 6 terms: n = 6. The first term is 65,930: a1 = 65,930. The
last term—that is, the sixth term—is 74,930: a6 = 74,930.
Sn =
n
1a + an2
2 1
S6 =
6
165,930 + 74,9302 = 31140,8602 = 422,580
2
Total adult residential community costs for your grandmother are predicted to be
$422,580. Because your grandmother’s assets are $500,000, she has enough to pay
for the facility for the six-year period.
Check Point
6
In Example 6, how much would it cost for the adult residential
community for a ten-year period beginning in 2009?
Exercise Set 10.2
Practice Exercises
29. a1 = - 20, d = - 4
30. a1 = - 70, d = - 5
In Exercises 1–14, write the first six terms of each arithmetic sequence.
31. a n = an - 1 + 3, a1 = 4
32. an = an - 1 + 5, a1 = 6
33. a n = an - 1 - 10, a1 = 30
34. an = an - 1 - 12, a1 = 24
1. a1 = 200, d = 20
2. a1 = 300, d = 50
3. a1 = - 7, d = 4
4. a1 = - 8, d = 5
5. a1 = 300, d = - 90
6. a1 = 200, d = - 60
7. a1 = 52 , d = - 12
8. a1 = 34 , d = - 14
9. an = an - 1 + 6, a1 = - 9
10. an = an - 1 + 4, a1 = - 7
11. an = an - 1 - 10, a1 = 30
12. an = an - 1 - 20, a1 = 50
35. Find the sum of the first 20 terms of the arithmetic sequence:
4, 10, 16, 22, Á .
36. Find the sum of the first 25 terms of the arithmetic sequence:
7, 19, 31, 43, Á .
37. Find the sum of the first 50 terms of the arithmetic sequence:
-10, - 6, -2, 2, Á .
13. an = an - 1 - 0.4, a1 = 1.6
14. an = an - 1 - 0.3, a1 = - 1.7
38. Find the sum of the first 50 terms of the arithmetic sequence:
-15, -9, - 3, 3, Á .
In Exercises 15–22, find the indicated term of the arithmetic sequence
with first term, a 1 , and common difference, d.
39. Find 1 + 2 + 3 + 4 + Á + 100, the sum of the first 100
natural numbers.
15. Find a6 when a1 = 13, d = 4.
16. Find a16 when a1 = 9, d = 2.
40. Find 2 + 4 + 6 + 8 + Á + 200, the sum of the first 100
positive even integers.
17. Find a50 when a1 = 7, d = 5.
41. Find the sum of the first 60 positive even integers.
18. Find a60 when a1 = 8, d = 6.
42. Find the sum of the first 80 positive even integers.
19. Find a200 when a1 = - 40, d = 5.
20. Find a150 when a1 = - 60, d = 5.
43. Find the sum of the even integers between 21 and 45.
44. Find the sum of the odd integers between 30 and 54.
21. Find a60 when a1 = 35, d = - 3.
22. Find a70 when a1 = - 32, d = 4.
In Exercises 23–34, write a formula for the general term (the nth
term) of each arithmetic sequence. Do not use a recursion formula.
Then use the formula for a n to find a20 , the 20th term of the sequence.
23. 1, 5, 9, 13, Á
24. 2, 7, 12, 17, Á
25. 7, 3, - 1, - 5, Á
26. 6, 1, -4, -9, Á
27. a1 = 9, d = 2
28. a1 = 6, d = 3
For Exercises 45–50, write out the first three terms and the last
term. Then use the formula for the sum of the first n terms of an
arithmetic sequence to find the indicated sum.
17
45. a 15i + 32
i=1
40
48. a 1- 2i + 62
i=1
20
46. a 16i - 42
i=1
100
49. a 4i
i=1
30
47. a 1 -3i + 52
i=1
50
50. a 1 -4i2
i=1
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Practice Plus
Use the graphs of the arithmetic sequences 5an6 and 5bn6 to solve
Exercises 51–58.
54. If 5bn6 is a finite sequence whose last term is 93, how many
terms does 5bn6 contain?
55. Find the difference between the sum of the first 14 terms of
5bn6 and the sum of the first 14 terms of 5an6.
56. Find the difference between the sum of the first 15 terms of
5bn6 and the sum of the first 15 terms of 5a n6.
57. Write a linear function f1x2 = mx + b, whose domain is the
set of positive integers, that represents 5a n6.
63. Company A pays $24,000 yearly with raises of $1600 per year.
Company B pays $28,000 yearly with raises of $1000 per year.
Which company will pay more in year 10? How much more?
64. Company A pays $23,000 yearly with raises of $1200 per year.
Company B pays $26,000 yearly with raises of $800 per year.
Which company will pay more in year 10? How much more?
In Exercises 65–67, we revisit the data from Chapter P showing the
average cost of tuition and fees at public and private four-year U.S.
colleges.
58. Write a linear function g1x2 = mx + b, whose domain is the
set of positive integers, that represents 5bn6.
Average Cost of Tuition and Fees at
Four-Year United States Colleges
Public Institutions
Use a system of two equations in two variables, a1 and d, to solve
Exercises 59–60.
60. Write a formula for the general term (the nth term) of the
arithmetic sequence whose third term, a3 , is 7 and whose
eighth term, a8 , is 17.
Tuition and Fees
59. Write a formula for the general term (the nth term) of the
arithmetic sequence whose second term, a2 , is 4 and whose
sixth term, a6 , is 16.
$20,000
$15,000
$10,000
$5,000
Application Exercises
The bar graphs show changes that have taken place in the United
States over time. Exercises 61–62 involve developing arithmetic
sequences that model the data.
Percentage of United
States Adults with
No Close Friends
Percentage of High School
Grades of Aⴙ, A, or Aⴚ
for College Freshmen
30%
24.6
15%
10%
10
5%
0
47.5
50%
Percent
Percent
20%
40%
30%
20%
17.6
10%
1985
2004
Year
Source: American Sociological Review
0
2004
2005
2006
2007
Ending Year in the School Year
Source: The College Board
65. a. Use the numbers shown in the bar graph to find the total
cost of tuition and fees at public colleges for a four-year
period from the school year ending in 2004 through the
school year ending in 2007.
b. The model
60%
25%
Private Institutions
$25,000
22,218
53. If 5a n6 is a finite sequence whose last term is -83, how many
terms does 5a n6 contain?
b. If trends shown by the model in part (a) continue, what
percentage of high school grades for college freshmen
will consist of A’s in 2018?
5836
52. Find a16 + b18 .
51. Find a14 + b12 .
a. Write a formula for the nth term of the arithmetic
sequence that models the percentage of high school
grades of A for college freshmen n years after 1967.
21,235
n
5491
1 2 3 4 5
20,082
n
62. In 1968, 17.6% of high school grades for college freshmen
consisted of A’s (A + , A, or A- ). On average, this has
increased by approximately 0.83% per year.
5132
1 2 3 4 5
b. If trends shown by the model in part (a) continue, what
percentage of Americans will have no close friends in
2011? Round to one decimal place.
13
11
9
7
5
3
1
−1
−3
−5
−7
19,710
13
11
9
7
5
3
1
−1
−3
−5
−7
bn
a. Write a formula for the nth term of the arithmetic
sequence that models the percentage of Americans with
no close friends n years after 1984.
4694
an
61. In 1985, 10% of Americans had no close friends. On average,
this has increased by approximately 0.77% per year.
1968
2004
Year
Source: www.grade-inflation.com
an = 379n + 4342
describes the cost of tuition and fees at public colleges in
academic year n, where n = 1 corresponds to the school
year ending in 2004, n = 2 to the school year ending in
2005, and so on. Use this model and the formula for Sn to
find the total cost of tuition and fees at public colleges for
a four-year period from the school year ending in 2004
through the school year ending in 2007. Does the model
underestimate or overestimate the actual sum you
obtained in part (a)? By how much?
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Section 10.2 Arithmetic Sequences
66. a. Use the numbers shown in the bar graph to find the total
cost of tuition and fees at private colleges for a four-year
period from the school year ending in 2004 through the
school year ending in 2007.
b. The model
an = 868n + 18,642
describes the cost of tuition and fees at private colleges
in academic year n, where n = 1 corresponds to the
school year ending in 2004, n = 2 to the school year ending in 2005, and so on. Use this model and the formula
for Sn to find the total cost of tuition and fees at private
colleges for a four-year period from the school year ending in 2004 through the school year ending in 2007. Does
the model underestimate or overestimate the actual sum
that you obtained in part (a)? By how much?
67. Use one of the models in Exercises 65–66 and the formula
for Sn to find the total cost of tuition and fees for your
undergraduate education. How well does the model describe
your anticipated costs?
68. A company offers a starting yearly salary of $33,000 with raises
of $2500 per year. Find the total salary over a ten-year period.
69. You are considering two job offers. Company A will start you
at $19,000 a year and guarantee a raise of $2600 per year.
Company B will start you at a higher salary, $27,000 a year, but
will only guarantee a raise of $1200 per year. Find the total
salary that each company will pay over a ten-year period.
Which company pays the greater total amount?
70. A theater has 30 seats in the first row, 32 seats in the second
row, increasing by 2 seats per row for a total of 26 rows. How
many seats are there in the theater?
71. A section in a stadium has 20 seats in the first row, 23 seats in
the second row, increasing by 3 seats each row for a total of
38 rows. How many seats are in this section of the stadium?
971
Critical Thinking Exercises
Make Sense? In Exercises 78–81, determine whether each
statement makes sense or does not make sense, and explain
your reasoning.
78. Rather than performing the addition, I used the formula
Sn = n2 1a1 + an2 to find the sum of the first thirty terms of
the sequence 2, 4, 8, 16, 32, Á .
79. I was able to find the sum of the first fifty terms of an arithmetic
sequence even though I did not identify every term.
80. The sequence for the number of seats per row in our movie
theater as the rows move toward the back is arithmetic
with d = 1 so people don’t block the view of those in the
row behind them.
81. Beginning at 6:45 A.M., a bus stops on my block every
23 minutes, so I used the formula for the nth term of an
arithmetic sequence to describe the stopping time for the
nth bus of the day.
82. In the sequence 21,700, 23,172, 24,644, 26,116, Á , which
term is 314,628?
83. A degree-day is a unit used to measure the fuel requirements
of buildings. By definition, each degree that the average
daily temperature is below 65°F is 1 degree-day. For
example, a temperature of 42°F constitutes 23 degree-days.
If the average temperature on January 1 was 42°F and fell
2°F for each subsequent day up to and including January 10,
how many degree-days are included from January 1 to
January 10?
84. Show that the sum of the first n positive odd integers,
1 + 3 + 5 + Á + 12n - 12,
is n2.
Writing in Mathematics
72. What is an arithmetic sequence? Give an example with your
explanation.
Preview Exercises
73. What is the common difference in an arithmetic sequence?
Exercises 85–87 will help you prepare for the material covered in
the next section.
74. Explain how to find the general term of an arithmetic sequence.
75. Explain how to find the sum of the first n terms of an arithmetic
sequence without having to add up all the terms.
Technology Exercises
76. Use the 冷SEQ 冷 (sequence) capability of a graphing utility and
the formula you obtained for an to verify the value you found
for a20 in any five exercises from Exercises 23–34.
77. Use the capability of a graphing utility to calculate the sum of a
sequence to verify any five of your answers to Exercises 45–50.
a2 a3 a4
85. Consider the sequence 1, - 2, 4, -8, 16, Á . Find , , ,
a1 a2 a3
a5
and . What do you observe?
a4
86. Consider the sequence whose nth term is an = 3 # 5n. Find
a5
a2 a3 a4
, , , and . What do you observe?
a1 a2 a3
a4
87. Use the formula an = a13n - 1 to find the 7th term of the
sequence 11, 33, 99, 297, Á .