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C. Series and Partial Sums

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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Section 9.1 Sequences and Series



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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9.2



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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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







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







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







Finding a Specified Term in an Arithmetic Sequence

Find the 24th term of the sequence 0.1, 0.4, 0.7, 1, p .



Solution







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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C. Series and Partial Sums

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