C. Series and Partial Sums
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CHAPTER 9 Additional Topics in Algebra
Finite Series
Given the sequence a1, a2, a3, a4, p , an, the sum of the terms is called a finite series
or partial sum and is denoted Sn:
Sn ϭ a1 ϩ a2 ϩ a3 ϩ p ϩ anϪ1 ϩ an
EXAMPLE 6
ᮣ
Computing a Partial Sum
Given an ϭ 2n, find the value of
Solution
ᮣ
a. S4
b. S7.
Since we eventually need the sum of the first seven terms, begin by writing out
these terms: 2, 4, 6, 8, 10, 12, and 14.
a. S4 ϭ a1 ϩ a2 ϩ a3 ϩ a4
b. S7 ϭ a1 ϩ a2 ϩ a3 ϩ a4 ϩ a5 ϩ a6 ϩ a7
ϭ2ϩ4ϩ6ϩ8
ϭ 2 ϩ 4 ϩ 6 ϩ 8 ϩ 10 ϩ 12 ϩ 14
ϭ 20
ϭ 56
Now try Exercises 51 through 56
Figure 9.12
ᮣ
Figure 9.13
There are several ways of computing a partial
sum using technology, with the most common
being (1) storing the sequence in a list then computing the sum of the list elements, or (2) computing
the sum of a sequence directly on the home screen
using the appropriate commands. On many calculators, the “sum(” option is in a MATH subSTAT
menu, accessed using 2nd
(LIST)
(MATH) 5:sum(. For the sum in Example 6(b), we
have u1n2 ϭ 2n, with option (1) demonstrated in Figure 9.12 and option (2) in
Figure 9.13.
C. You’ve just seen how
we can find the partial sum
of a series
D. Summation Notation
When the general term of a sequence is known, the Greek letter sigma © can be used to
write the related series as a formula. For instance, to indicate the sum of the first four term
4
of an ϭ 3n ϩ 2, we write
͚ 13i ϩ 22 with this notation indicating we are to compute
iϭ1
the sum of all terms generated as i cycles from 1 through 4. This result is called summation or sigma notation and the letter i is called the index of summation. The letters
j, k, l, and m are also used as index numbers, and the summation need not start at 1.
EXAMPLE 7
ᮣ
Computing a Partial Sum
Compute each sum:
4
6
1
c.
1Ϫ12 kk2
iϭ1
jϭ1 j
kϭ3
d. Check each sum using a graphing calculator.
a.
͚
5
13i ϩ 22
b.
͚
͚
4
Solution
ᮣ
a.
͚ 13i ϩ 22 ϭ 13 # 1 ϩ 22 ϩ 13 # 2 ϩ 22 ϩ 13 # 3 ϩ 22 ϩ 13 # 4 ϩ 22
iϭ1
5
b.
ϭ 5 ϩ 8 ϩ 11 ϩ 14 ϭ 38
1
1
1
1
1
1
͚j ϭ1ϩ2ϩ3ϩ4ϩ5
jϭ1
ϭ
60
30
20
15
12
137
ϩ
ϩ
ϩ
ϩ
ϭ
60
60
60
60
60
60
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6
c.
͚ 1Ϫ12 k
k 2
767
ϭ 1Ϫ12 3 # 32 ϩ 1Ϫ12 4 # 42 ϩ 1Ϫ12 5 # 52 ϩ 1Ϫ12 6 # 62
ϭ Ϫ9 ϩ 16 ϩ 1Ϫ252 ϩ 36 ϭ 18
d. Begin by entering the functions in parts (a), (b), and (c) as u(n), v(n), and w(n)
respectively on the Y= screen (Figure 9.14). Note that while different indices
are used in this example, all are entered into the calculator using the variable n.
The results are shown in Figures 9.15 and 9.16.
kϭ3
Figure 9.14
Figure 9.16
Figure 9.15
Now try Exercises 57 through 68
ᮣ
If a definite pattern is noted in a given series expansion, this process can be
reversed, with the expanded form being expressed in summation notation using the
nth term.
EXAMPLE 8
ᮣ
Writing a Sum in Sigma Notation
Write each of the following sums in summation (sigma) notation.
a. 1 ϩ 3 ϩ 5 ϩ 7 ϩ 9
b. 6 ϩ 9 ϩ 12 ϩ 15 ϩ p
Solution
ᮣ
a. The series has five terms and each term is an odd number, or 1 less than a
5
multiple of 2. The general term is an ϭ 2n Ϫ 1, and the series is
͚ 12n Ϫ 12.
nϭ1
WORTHY OF NOTE
b. The raised ellipsis “ p ” indicates the sum continues infinitely. Since the terms
are multiples of 3, we identify the general term as an ϭ 3n, while noting the
series starts at n ϭ 2 (instead of n ϭ 1). Since the sum continues indefinitely,
we use the infinity symbol q as the “ending” value in sigma notation. The
q
By varying the function given and/or
where the sum begins, more than
one acceptable form is possible.
series is
͚ 3n.
nϭ2
For Example 8(b)
13 ϩ 3k2
kϭ1
also works.
q
͚
Now try Exercises 69 through 78
ᮣ
Since the commutative and associative laws hold for the addition of real numbers,
summations have the following properties:
Properties of Summation
Given any real number c and natural number n,
n
(I)
͚ c ϭ cn
iϭ1
If you add a constant c “n” times the result is cn.
n
(II)
͚
n
cai ϭ c
iϭ1
͚a
i
iϭ1
A constant can be factored out of a sum.
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n
(III)
n
n
͚ 1a Ϯ b 2 ϭ ͚ a Ϯ ͚ b
i
i
i
iϭ1
i
iϭ1
iϭ1
A summation can be distributed to two (or more) sequences.
m
(IV)
͚
n
iϭ1
n
͚
ai ϩ
ai ϭ
iϭmϩ1
͚a; 1 Յ m 6 n
i
iϭ1
A summation is cumulative and can be written as a sum of smaller parts.
The verification of property II depends solely on the distributive property.
n
Proof:
͚ ca ϭ ca
i
iϭ1
1
ϩ ca2 ϩ ca3 ϩ p ϩ can
expand sum
ϭ c1a1 ϩ a2 ϩ a3 ϩ p ϩ an 2
factor out c
ϭc
write series in summation form
n
͚a
i
iϭ1
The verifications of properties III and IV simply use the commutative and associative properties. You are asked to prove property III in Exercise 94.
EXAMPLE 9
ᮣ
Computing a Sum Using Summation Properties
4
Recompute the sum
͚ 13i ϩ 22 from Example 7(a) using summation properties.
iϭ1
4
Solution
ᮣ
͚
iϭ1
13i ϩ 22 ϭ
4
͚
4
͚2
property III
͚i ϩ ͚2
property II
3i ϩ
iϭ1
4
ϭ3
iϭ1
iϭ1
4
iϭ1
ϭ 31102 ϩ 2142
ϭ 38
D. You’ve just seen how
we can use summation
notation to write and
evaluate series
1 ϩ 2 ϩ 3 ϩ 4 ϭ 10; property I
result
Now try Exercises 79 through 82
ᮣ
E. Applications of Sequences
To solve applications of sequences, (1) identify where the sequence begins (the initial
term), (2) write out the first few terms to help identify the nth term, and (3) decide on
an appropriate approach or strategy.
EXAMPLE 10
ᮣ
Solving an Application — Accumulation of Stock
Hydra already owned 1420 shares of stock when her company began offering
employees the opportunity to purchase 175 discounted shares per year. If she made
no purchases other than these discounted shares each year, how many shares will
she have 9 yr later? If this continued for the 25 yr she will work for the company,
how many shares will she have at retirement?
Solution
ᮣ
To begin, it helps to simply write out the first few terms of the sequence. Since she
already had 1420 shares before the company made this offer, we let a0 ϭ 1420 be
the inaugural element, showing a1 ϭ 1595 (after 1 yr, she owns 1420 ϩ 175 ϭ
1595 shares). The first few terms are 1595, 1770, 1945, 2120, and so on. This
supports a general term of an ϭ 1595 ϩ 1751n Ϫ 12.
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After 9 years
After 25 years
a9 ϭ 1595 ϩ 175182
ϭ 2995
a25 ϭ 1595 ϩ 1751242
ϭ 5795
769
After 9 yr she would have 2995 shares. Upon retirement she would own 5795
shares of company stock.
Now try Exercises 85 through 90
ᮣ
Surprisingly, sequences and series have a number of other interesting properties,
applications, and mathematical connections. For instance, some of the most celebrated
numbers in mathematics can be approximated using a series, as demonstrated in
Example 11.
EXAMPLE 11
ᮣ
Use a calculator to find the partial sums S4, S8, and S12 for the sequences given. If
any sum seems to be approaching a fixed number, name that number.
a. an ϭ
Solution
ᮣ
1
n!
b. an ϭ
1
3n
1
as u(n) on the Y= screen. Using the “sum(” and “seq(”
n!
commands as before produces the results shown in Figures 9.17 through 9.19,
where it appears the sum becomes very close to e Ϫ 1 for larger values of n.
a. Begin by entering
Figure 9.17
Figure 9.18
Figure 9.19
1
as v(n) on the Y= screen, we once again compute the sums
3n
indicated as shown in Figures 9.20 through 9.22. It appears the sum becomes
1
very close to for larger values of n.
2
Figure 9.20
Figure 9.22
Figure 9.21
b. After entering
E. You’ve just seen how
we can use sequences to
solve applications
Now try Exercises 91 and 92
ᮣ
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CHAPTER 9 Additional Topics in Algebra
9.1 EXERCISES
ᮣ
CONCEPTS AND VOCABULARY
Fill in the blank with the appropriate word or phrase. Carefully reread the section if needed.
1. A sequence is a(n)
specific
.
ᮣ
of numbers listed in a
2. A series is the
given sequence.
of the numbers from a
3. A sequence that uses the preceding term(s) to generate
those that follow is called a
sequence.
4. The notation n! represents the
natural number n, with all those
5. Describe the characteristics of a recursive sequence
and give one example.
6. Describe the characteristics of an alternating
sequence and give one example.
of the
n.
DEVELOPING YOUR SKILLS
Find the first four terms, then find the 8th and 12th
term for each nth term given.
1 n
29. an ϭ a1 ϩ b ; a10
n
1 n
30. an ϭ an ϩ b ; a9
n
7. an ϭ 2n Ϫ 1
8. an ϭ 2n ϩ 3
9. an ϭ 3n Ϫ 3
10. an ϭ 2n3 Ϫ 12
31. an ϭ
1
; a4
n12n ϩ 12
12. an ϭ
32. an ϭ
1
; a5
12n Ϫ 12 12n ϩ 12
2
11. an ϭ 1Ϫ12 nn
13. an ϭ
n
nϩ1
1 n
15. an ϭ a b
2
17. an ϭ
19. an ϭ
1
n
1 n
14. an ϭ a1 ϩ b
n
2 n
16. an ϭ a b
3
18. an ϭ
1Ϫ12 n
n1n ϩ 12
21. an ϭ 1Ϫ12 n2n
1Ϫ12 n
n
20. an ϭ
1
n2
1Ϫ12 nϩ1
2n Ϫ 1
2
22. an ϭ 1Ϫ12 n2Ϫn
Use a calculator to: (a) find the indicated term for each
sequence, and (b) generate the first five terms of each
sequence and store the results in a list. Use fractions if
possible; round to tenths when necessary.
23. an ϭ n2 Ϫ 2; a9
24. an ϭ 1n Ϫ 22 2; a9
25. an ϭ
26. an ϭ
1Ϫ12 nϩ1
; a5
n
1 nϪ1
27. an ϭ 2a b ; a7
2
1Ϫ12 nϩ1
2n Ϫ 1
; a5
1 nϪ1
28. an ϭ 3a b ; a7
3
Find the first five terms of each recursive sequence.
33. e
a1 ϭ 2
an ϭ 5anϪ1 Ϫ 3
34. e
a1 ϭ 3
an ϭ 2anϪ1 Ϫ 3
35. e
a1 ϭ Ϫ1
an ϭ 1anϪ1 2 2 ϩ 3
36. e
a1 ϭ Ϫ2
an ϭ anϪ1 Ϫ 16
38. e
c1 ϭ 1, c2 ϭ 2
cn ϭ cnϪ1 ϩ 1cnϪ2 2 2
c1 ϭ 64, c2 ϭ 32
37. •
cnϪ2 Ϫ cnϪ1
cn ϭ
2
Simplify each factorial expression.
39.
8!
5!
40.
12!
10!
41.
9!
7!2!
42.
6!
3!3!
43.
8!
2!6!
44.
10!
3!7!
Write out the first four terms in each sequence.
45. an ϭ
n!
1n ϩ 12!
46. an ϭ
n!
1n ϩ 32!
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1n ϩ 12!
47. an ϭ
48. an ϭ
13n2!
nn
n!
49. an ϭ
50. an ϭ
1n ϩ 32!
70. a. Ϫ1 ϩ 4 Ϫ 9 ϩ 16 Ϫ 25 ϩ 36
b. 1 Ϫ 8 ϩ 27 Ϫ 64 ϩ 125 Ϫ 216
71. a. 1 ϩ 3 ϩ 5 ϩ 7 ϩ 9 ϩ 11 ϩ p
12n2!
2n
n!
1
1
1
1
1
ϩ ϩ ϩ
ϩ
ϩp
2
4
8
16
32
72. a. 0.1 ϩ 0.01 ϩ 0.001 ϩ 0.0001 ϩ p
1
1
1
1
1
b. 1 ϩ ϩ ϩ
ϩ
ϩ
ϩp
2
6
24
120
720
b. 1 ϩ
Find the indicated partial sum for each sequence.
51. an ϭ n; S5
52. an ϭ n2; S7
53. an ϭ 2n Ϫ 1; S8
54. an ϭ 3n Ϫ 1; S6
1
55. an ϭ ; S5
n
56. an ϭ
n
; S4
nϩ1
For the given general term an, write the indicated sum
using sigma notation.
Expand and evaluate each series. Verify results using a
graphing calculator.
4
57.
60.
͚ 13i Ϫ 52
5
58.
59.
͚ 12k Ϫ 32
2
iϭ1
iϭ1
kϭ1
5
7
5
͚ 1k
2
ϩ 12
61.
kϭ1
4
͚
2
7
j
i
63.
iϭ1 2
66.
5
͚ 12i Ϫ 32
k
j
62.
kϭ1
64.
͚i
iϭ2
67.
͚ 1Ϫ12 2
k k
kϭ1
4
8
͚2
jϭ3
͚ 1Ϫ12 k
7
2
65.
͚ 2j
jϭ3
1Ϫ12 k
6
͚ k1k Ϫ 22
68.
kϭ3
1Ϫ12 kϩ1
͚k
kϭ2
2
Ϫ1
Write each sum using sigma notation. Answers are not
necessarily unique.
69. a. 4 ϩ 8 ϩ 12 ϩ 16 ϩ 20
b. 5 ϩ 10 ϩ 15 ϩ 20 ϩ 25
73. an ϭ n ϩ 3; S5
74. an ϭ
n2 ϩ 1
; S4
nϩ1
75. an ϭ
n2
; third partial sum
3
76. an ϭ 2n Ϫ 1; sixth partial sum
77. an ϭ
Compute each sum by applying properties of summation.
5
79.
͚
iϭ1
80.
13k2 ϩ k2
82.
4
81.
͚
6
14i Ϫ 52
͚ 13 ϩ 2i2
iϭ1
4
͚ 12k
3
ϩ 52
kϭ1
WORKING WITH FORMULAS
83. Sum of an ؍3n ؊ 2: Sn ؍
n13n ؊ 12
2
The sum of the first n terms of the sequence
defined by an ϭ 3n Ϫ 2 ϭ 1, 4, 7, 10, p ,
13n Ϫ 22, p is given by the formula shown. Find
S5 using the formula, then verify by direct
calculation.
ᮣ
n
; sum for n ϭ 3 to 7
2n
78. an ϭ n2; sum for n ϭ 2 to 6
kϭ1
ᮣ
771
n13n ؉ 12
2
The sum of the first n terms of the sequence defined
by an ϭ 3n Ϫ 1 ϭ 2, 5, 8, 11, p , 13n Ϫ 12, p is
given by the formula shown. Find S8 using the
formula, then verify by direct calculation. Observing
the formulas from Exercises 83 and 84, can you
now state the sum formula for an ϭ 3n Ϫ 0?
84. Sum of an ؍3n ؊ 1: Sn ؍
APPLICATIONS
Use the information given in each exercise to determine the nth term an for the sequence described. Then use the nth
term to list the specified number of terms.
85. Wage increases: Latisha gets $7.25 an hour for filling candy machines for Archtown Vending. Each year
she receives a $0.50 hourly raise. List Latisha’s hourly wage for the first 5 yr. How much will she make in the
fifth year if she works 8 hr per day for 240 working days?
86. Average birth weight: The average birth weight of a certain animal species is 900 g, with the baby gaining
125 g each day for the first 10 days. List the infant’s weight for the first 10 days. How much does the infant
weigh on the 10th day?
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87. Blue-book value: Steve’s car has a blue-book
value of $6000. Each year it loses 20% of its value
(its value each year is 80% of the year before). List
the value of Steve’s car for the next 5 yr.
(Hint: a0 ϭ 6000.)
88. Effects of inflation: Suppose inflation (an increase
in value) will average 4% for the next 5 yr. List the
growing cost (year by year) of a DVD that costs
$15 right now. (Hint: a0 ϭ 15.)
89. Stocking a lake: A local fishery stocks a large
lake with 1500 bass and then adds an additional
100 mature bass per month until the lake nears
maximum capacity. If the bass population grows
at a rate of 5% per month through natural
reproduction, the number of bass in the pond
after n months is given by the recursive sequence
b0 ϭ 1500, bn ϭ 1.05bnϪ1 ϩ 100. How many
bass will be in the lake after 6 months?
ᮣ
9–12
CHAPTER 9 Additional Topics in Algebra
90. Species preservation: The Interior Department
introduces 50 wolves (male and female) into a
large wildlife area in an effort to preserve the
species. Each year about 12 additional adult
wolves are added from capture and relocation
programs. If the wolf population grows at a rate
of 10% per year through natural reproduction,
the number of wolves in the area after n years
is given by the recursive sequence
w0 ϭ 50, wn ϭ 1.10wnϪ1 ϩ 12. How many wolves
are in the wildlife area after 6 years?
Use your calculator to find the partial sums for
n ؍4, n ؍8, and n ؍12 for the summations given,
and attempt to name the number the summation
approximates:
n
91.
2k ϩ 3k
6k
kϭ1
n
1
k
kϭ1 2
͚
92.
͚
EXTENDING THE CONCEPT
93. Verify that a constant can be factored out of a sum.
That is, verify that the following statement is true:
n
͚
n
caj ϭ c
jϭ1
͚a
j
jϭ1
94. Verify that a summation may be distributed to two
(or more) sequences. That is, verify that the
following statement is true:
n
͚
iϭ1
1ai Ϯ bi 2 ϭ
n
͚
n
ai Ϯ
iϭ1
͚b .
i
iϭ1
Regarding Exercises 91 and 92, sometimes a series will approach a fixed number very slowly, and many more terms
must be added before this value is recognized. Use your graphing calculator to compute the sums S10, S25, and S50 for
the following sequences to see if you can recognize the number. Add more terms if necessary.
95. an ϭ
ᮣ
1
n1n ϩ 12 1n ϩ 22
96. an ϭ
1
12n Ϫ 12 12n ϩ 12
MAINTAINING YOUR SKILLS
1
97. (5.3) Write log381
ϭ Ϫx in exponential form, then
solve by equating bases.
98. (3.6) Set up the difference quotient for f 1x2 ϭ 1x,
then rationalize the numerator.
99. (8.4) Solve the nonlinear system. e
x2 ϩ y2 ϭ 9
9y2 Ϫ 4x2 ϭ 16
100. (7.3) Solve the system using a matrix equation.
25x ϩ y Ϫ 2z ϭ Ϫ14
• 2x Ϫ y ϩ z ϭ 40
Ϫ7x ϩ 3y Ϫ z ϭ Ϫ13
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Arithmetic Sequences
LEARNING OBJECTIVES
Similar to the way polynomials fall into certain groups or families (linear, quadratic,
cubic, etc.), sequences and series with common characteristics are likewise grouped. In
this section, we focus on sequences where each successive term is generated by adding
a constant value, as in the sequence 1, 8, 15, 22, 29, p , where 7 is added to a given
term in order to produce the next term.
In Section 9.2 you will see
how we can:
A. Identify an arithmetic
sequence and its common
difference
B. Find the n th term of an
arithmetic sequence
C. Find the n th partial sum
of an arithmetic sequence
D. Solve applications
involving arithmetic
sequences
A. Identifying an Arithmetic Sequence and Finding
the Common Difference
An arithmetic sequence is one where each successive term is found by adding a fixed
constant to the preceding term. For instance 3, 7, 11, 15, p is an arithmetic sequence,
since adding 4 to any given term produces the next term. This also means if you take
the difference of any two consecutive terms, the result will be 4 and in fact, 4 is called
the common difference d for this sequence. Using the notation developed earlier, we
can write d ϭ akϩ1 Ϫ ak, where ak represents any term of the sequence and akϩ1 represents the term that follows ak.
Arithmetic Sequences
Given a sequence a1, a2, a3, p , ak, akϩ1, p , an, where k, n ʦ ގand k 6 n,
if there exists a common difference d such that akϩ1 Ϫ ak ϭ d for all k,
then the sequence is an arithmetic sequence.
The difference of successive terms can be rewritten as akϩ1 ϭ ak ϩ d (for k Ն 12
to highlight that each following term is found by adding d to the previous term.
EXAMPLE 1
ᮣ
Identifying an Arithmetic Sequence
Determine if the given sequence is arithmetic. If yes, name the common difference.
If not, try to determine the pattern that forms the sequence.
77 29
a. 2, 5, 8, 11, p
b. 12, 56, 13
12 , 60 , 20 , p
Solution
ᮣ
a. Begin by looking for a common difference d ϭ akϩ1 Ϫ ak. Checking each pair
of consecutive terms we have
5Ϫ2ϭ3
8Ϫ5ϭ3
11 Ϫ 8 ϭ 3 and so on.
This is an arithmetic sequence with common difference d ϭ 3.
b. Checking each pair of consecutive terms yields
5
1
5
3
Ϫ ϭ Ϫ
6
2
6
6
2
1
ϭ ϭ
6
3
13
5
13
10
Ϫ ϭ
Ϫ
12
6
12
12
3
1
ϭ
ϭ
12
4
13
77
65
77
Ϫ
ϭ
Ϫ
60
12
60
60
12
1
ϭ
ϭ
60
5
Since the difference is not constant, this is not an arithmetic sequence. It
appears the sequence is formed by adding 1k to each previous term, for natural
numbers k.
Now try Exercises 7 through 18
9–13
ᮣ
773
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CHAPTER 9 Additional Topics in Algebra
EXAMPLE 2
ᮣ
Writing the First k Terms of an Arithmetic Sequence
Write the first five terms of the arithmetic sequence, given the first term a1 and the
common difference d.
a. a1 ϭ 12 and d ϭ Ϫ4
b. a1 ϭ 12 and d ϭ 13
Solution
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A. You’ve just seen how
we can identify an arithmetic
sequence and its common
difference
a. a1 ϭ 12 and d ϭ Ϫ4. Starting at a1 ϭ 12, add Ϫ4 to each new term to
generate the sequence: 12, 8, 4, 0, Ϫ4.
b. a1 ϭ 12 and d ϭ 13. Starting at a1 ϭ 12 and adding 13 to each new term will
generate the sequence: 12, 56, 76, 32, 11
6 . Note that since the common denominator is
6, terms of the sequence can quickly be found by adding 13 ϭ 26 to the previous
term and reducing if possible.
Now try Exercises 19 through 30
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B. Finding the n th Term of an Arithmetic Sequence
If the values a1 and d from an arithmetic sequence are known, we could generate the
terms of the sequence by adding multiples of d to the first term, instead of adding d to
each new term. For example, we can generate the sequence 3, 8, 13, 18, 23 by adding
multiples of 5 to the first term a1 ϭ 3:
3 ϭ 3 ϩ 1025
a1 ϭ a1 ϩ 0d
13 ϭ 3 ϩ 1225
a3 ϭ a1 ϩ 2d
8 ϭ 3 ϩ 1125
a2 ϭ a1 ϩ 1d
18 ϭ 3 ϩ 1325
a4 ϭ a1 ϩ 3d
23 ϭ 3 ϩ 1425
current term
initial
term
S
S
a5 ϭ a1 ϩ 4d
coefficient of common
difference
It’s helpful to note the coefficient of d is 1 less than the subscript of the current
term (as shown): 5 Ϫ 1 ϭ 4. This observation leads us to a formula for the nth term.
The n th Term of an Arithmetic Sequence
The nth term of an arithmetic sequence is given by
an ϭ a1 ϩ 1n Ϫ 12d
where d is the common difference.
EXAMPLE 3
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Finding a Specified Term in an Arithmetic Sequence
Find the 24th term of the sequence 0.1, 0.4, 0.7, 1, p .
Solution
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Instead of creating all terms up to the 24th, we determine the constant d and use
the nth term formula. By inspection we note a1 ϭ 0.1 and d ϭ 0.3.
an ϭ a1 ϩ 1n Ϫ 12d
ϭ 0.1 ϩ 1n Ϫ 120.3
ϭ 0.1 ϩ 0.3n Ϫ 0.3
ϭ 0.3n Ϫ 0.2
n th term formula
substitute 0.1 for a1 and 0.3 for d
eliminate parentheses
simplify
To find the 24th term we substitute 24 for n:
a24 ϭ 0.31242 Ϫ 0.2
ϭ 7.0
substitute 24 for n
result
Now try Exercises 31 through 42
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