f
9i
d-'
ej
Gsilil#q rAq il ?ln b
168 NumberRepresentation
andGlculation. Chapter4
BLITZER
BONUS
Numbers
andBirdBrains
Numerarsin the Mayan systemare expressedvertically.The prace
value at ths
.
bottom of the columnis 1_
Write
a
asa Hindu-Arabic
numeral.
Birds have largg well-developed
brains and are more intelligent
than is suggestedby the slur
"bird brain." Parbkeetscan leam
to count to seven.Theyhave
been taught to identify a box of
food by countingthe numberof
smallobjectsin front of the box.
Somespeciesof birds can tell
the differencebetweentwo and
three.If a nest containsfour
eggsand one is taken, the bird
will stayin the nest to protect
the remainingthree eggs However,iftwo ofthe four eggsare
taken, the bird recognizesthat
only two remain and will desert
the nest,leavingthe remaining
two eggsunprotected.
Birds easilymastercomplex
countingproblemsThesenseof
more and lessthat led to the developmentof numeiationsystemsis
not limited to the humanqpecies
tOLlmON The givenMrIp numeralhasfour places.From top to
bottom,t[e
placevaluesare72a0,360,20,
andL.Represent
thenumeralin eachrowasa familiar Hindu-Arabicnumeralushq"rlbt-"+.2.t'tuttipty eachHindu-Arabi"
ou*.rJiy
its respective
placevalue.Thenhnd
thesumof thereproducts.
Mayan
numeral
Hindu-Arabic
numeral
:=I+X
Place
value
7200
9=0x
=tx
3ffi
?n
=tLx
100,800
0
1.40
12
100.952
The sum,onthe right indicatesthat the given Mayan numeral
is 100,952whenwritten asa Hindu-Arabic numeral.
Write
as a Hindu-Arabic numeral.
Exercise
Set4.1
PracticeExercises
n exercx@varuate theexpression
l, 52
Y2.
q14
6,
6. 24
In Exercises
Arabic
3.23
4. 43
23. (7 x 10t)+ (3 x 1)
7. 10s
8. 106
24. (9 x 101)+ (4 x 1)
.-\
In Exercise(9-22,)trite
eachHindu-Arabicnumeralin expanded
\--l
fornt.
9. 36
13.703
10. 65
71,249
14.902
15. 4856
17.3070
18. 9007
21 230,W7,004
expresseach expandedform as a Hindu.
U. 698
t6.574g
79.34,569 20, 67,943
22. 909.N6.070
2s. (3 x 102)+ (8 x 101)+ (s x 1)
26. (7 x 102)+ (5 x 101)+ (3 x 1)
27. (s x 10s)+ (2 x 104)+ (8 x 103)+ (z x 102)
+(4x101)+(3x1)
28. (7 x 106)+ (4 x 105)+ (2 x 10{)+ (3 x 103)
+ (1 x 102)+ (9 x 101)+ (6 x 1)
Section4.1 o Our Hindu-A,rabic
Systemrnd EarlyPositional
Systems 169
ue at the
+ (0 x 102)+ (0 x 101)+ (2 x i)
2s.(7 x103)
+ (0 x 1d) + (0 x 1d) + (4 x 101)
30.(9x 104)
+(5x1)
+ (2 x ld) + (7 x 1)
tr.(6x 108)
x
+ (4 x 1)
3?.p x108)+ (5 104)
a3340,ltse Table4.1 on page166to write eachBaby-
tn Exercis(s
numeral'
ionion nurloqglastl Hindu-Arabic
<<<vv
3
4
.
vvv
33. < <
ttom, the
s a familmeral by
58. Describeone way that the Babyloniansystemis similar to
the Hindu-Arabic system and one way that it is different
from the Hindu-Arabic system.
59. Describeone way that the Mayan systemis similar to the
Hindu-Arabic system and one way that it is different from
the Hindu-Arabic system.
60. Research activity W:'Jttea report on the history of the
Hindu-Arabic system of numeration. Useful references
include history of mathematics bookg encyclopediag and
theWorldWideWeb.
3 5 .< < v v v
36.
3 7 .v v v < v v v v v
38.
<V <<VV
61. Write v <vv
39.<v <v <v <v
'f0.
<< <VV <VV
62. Write "
Critical
Thinking
Exercises
.__-_,
<v asaMayannumeral.
asa Babylonian numeral.
t\
ln Exercbe\41-50,tyseTable 4.2 on page 167 to write each
Mayannumels,Lgi'/ H indu-Arab ic numeral.
':::
43. =
4L ::!
U. '-:::
4L
45.:::-
'
.a:
4.
a€
--:47. ';
63. Use Babylonian numeralsto write the numeral that precedesand the numeral that follows
<v
<<<<<vvvvvvvvv
.E "'
Technology
Exercises
-:l::9Q
--L
49.:
-L:-
I
5 0 .:
In Exercises6447, we a calculator to evaluateeachexpression.
64.4s
65. 54
ffi.2e
67. 843
\J
hen writ-
I
Group Exercises
ApplicationExercises
F
&
Exerciie51 contains a Mayan numeral. Exercise52 contains a
Babyloniannumeral. Write.each numeral as a Hind.u-Arabic
numeral.Thenumeral that resultsrepresentsa year that was not
tmportantto the Mayans or the Babylonians,but b important to
us.Identifi the historical event that took place in that year If
necessary,
usean appropriate reference.
5 1 ." "
52. <<vvvv <<<<<vv
-r
+
Writingin Mathematics
53. Describethe differencebetweena number and a numeral.
a Hindu-
54. Explainhow to evaluate73.
55. What is the basein our Hindu-Arabic numerationsystem?
What are the digitsin the system?
56. Why is a symbolfor zero neededin a positionalsystem?
57. Explainhow to write a Hindu-Arabic numeralin expanded
form.
)
+(3x1)
)
+(6x1)
68. Your group task is to create an original positional numeration systemthat is different from the three systemsdiscussed
in this section.
a. Constructa table showingyour numeralsand the correspondingHindu-Arabic numerals.
b. Explain how to representnumbersin your system,and
express a three-digit and a four-digit Hindu-Arabic
numeral in your system.
69. For many individuals around the world who are blind or
visually impaired, Braille numeralsare used to represent
numbers.Braille numeralscan be detectedand interpreted
by touch.A group of two or three studentsshouldresearch
how Braille numeralsare formed. Many public and private
organizationsand schoolsprovide materials'andinformation for the blind and visuallyimpaired.Usingyour library,
resourceson the World Wide Web, or local organizations,
investigatethe basicelementsfor Braille numerals,including the two-dot by three-dotcell, as well as the dot symbol
used to precede a numeral. Present the results of your
researchto the entire classand bring examplesof Braille
numerals for each student to intemret bv hand with no
visualclues
Section4.2 r NumberBasesin PositionalSystcms 115
to determeral.
tnrciseSet 4'2
iurticuE14"'
n F.xercis$l-I8)convert the numeral to a numeral in baseten.
*r.""*,n_"
\-/
3. 52.ir1r1
2.346u.
rsto show
de 299by
4, 67ei*,t
5' 1321ou'
6' 3216"
7, 1011t*o
8' 1101t*o
9' 2035'i*
11' ?0355"ig11
12' 4t502sb
lj.2@6"ir1""o
14. 31046x""n
15. 110101h,o
16. 101101t*o
17' ACE5.;'1""'
18' EDFful*o
L0.20'73m\e
46. Describehowtochangeabasetennumeraltoanumeralin
anotherbase.
Thinking
Exerciser
Critical
InExercisu474|,writeintheindicatedbasethecountingnumbers that precedeand follow the number expressedby the given
numeraL
47. gggo6u
48. EC5ri*t""n
ln Exercis$ L9-28fuenmlly convert eachbaseten numeral to a
sumeralinfu'gjtcn base
i of 8, and
&
49. Arrange from smallest to largest:
11111011,t*o,3A6s"p",673"g,1.
19. 7 to basefive
20. 9 to basefive
2L lltobaseseven
22. 12tobaseseven
E.2tobasetwo
A- 3to basetwo
Technology Exercises
25. 13to basefour
26. L9 to basefour
In Exercbes 50-54,use the scientiftc calculator in Windows (or
six
2?.37tobase
six
8. 25tobase
fl:::::::,f:#::;'n:m::i;{:;::::tr:;;";:';"^,)'
If you are unsure how to convert to a dffirent baseon your calculator, consult the owner's manual.
51. 1''1'l,otodecimal
50.45tooctal
ln Exerclsu2940, use divisiors to convert eachbaseten numeraltoanumeralinthegivenbase.
3{l' 85 to baseseven
29' 87to basefive
3f i08 to basefour
32' 1nnto basefour
33.l9tobasetwo
34,23to basetwo
35' 138to basethrge
36' 129to basethree
37' 386to basesix
38' 428to basenine
39. 15ggto baseseven
40. 1346tobaseeight
to hexadecimal 53. 567 to binary
sz. 10010r"two
to hexadecimar
54. 333g1"h1
Group Exercises
'
risionsby
4tsro,ro,
'
of?
O;orie1.
,,
ffigver
write the
&
Application Exercises
Usea proceduresimilar to the one usedin Exercises2940 to
solveExercbes
4143.
41. Change153daysto weeksand days.
The foltowing topicsare appropriatefor either individual or
be given to the classon
group
'the researchproiectsA report should
researchedtopic. IJseful referencesinclude history of mathe'
maticsbooks,books whosepurposeis to excitethe readerabout
and.theworld wide web.
encycloped.ias,
mathematics,
55' Societiesthat Use Numeration Systemswith BasesOther
ThanTen
56, The Use of Fingersto RepresentNumbers
42. Change273hourstodaysandhours.
vsand hours.
57. Applications ofBases OtherThanTen
43. Change$8.79to quarters,nickels,and pennies.
53: Binary Octal,HexadecimalBasesand Computers
Witing in Mathematics
44. Explain how to determinethe place valuesfor a four-digit
numeral in basesix.
59. Negative Bases(See "Numeration Systemswith Unusual
Bases,"by David Ballow in The MathematicsTeacher,May
L974'pp'4134r4')
60. Babylonian and Mayan Civilizations andTheir Contributions
4.3 r Computation
in Positional
Systems181
Section
Letls check the quotient by converting to base ten: 31ou,: 3,2226u, = 42, and
1A
= L4.Because
workis correct.
32rou,
3fr,out
&
Use Table 4.5,showingproductsin basefour, to perform the following
division:
: show-
&
2rou,)ii2[.
Set4.3
Exercise
liULfl PLICAflON:BA5EFIVE
Practice
l*ereiles
on:
in base
:olumn
rductis
ln Exerc{p Ll2, )d
23r"\--/
l.
* 13rou
4.
7.
10.
8.
645r"""n
+ 324se"e\
lL
102ltu"s
* 201Ltte"
tn E rrrirffz\btract
/
13. 32}'.igits of
2.
5.
101r*o
+ 1lt*o
X
n the ndicated base.
31rou.
* 22uw
3.
342fi,e
* 413n"e
6.
9.
632s"""n
+ 564sven
n.
L4632*"uo
+ 5604s"en
in theindicatedbase.
15.
M.
2rro,,
- I?to-
- 13tou,
1
0
2
4
3
0
J
4
0
4
6784n1n"
* ?865nioe
53Br;*1""n
* 694sixteen
20.
462"isht
_ 177eieht
2L.
1001t*o
- L11two
23..
12006,""
- 1012tu""
24.
4C6r1"q""n
- 1.98ri"t""n
X
31.
X'
34.
X
3fou,
e3.isnt
5435;1
4eient
X .. 5rt"
'-
^v l
rtwo
243"""
X
33.
712nin
483nin"
11,*o
30.
5r"u"n
6nin"
In Exercises35-38,use the multiplication tablesshown below
andin the next column to divide in the intlicated base.
BAIEFOUR
lltl[TlPLlCArlON:
3
0
I
0
0
0
0
0
I
0
I
2
3
)
0
2
10
t2
3
0
J
t2
2l
X
LL
13
11
14
22
13
n
JI
36. 2rou,)]Ii[
37.3nu"w;
38.4,i""Ir4;
Writingin Mathematics
39. Describehow to add two numbersin a baseother than ten.
How do you expressand record the sum of numbersin a
column if that sum exceedsthe base?
40. Describehow to subtracttwo numbersin a baseother than
ten. How do you subtract a llrrger number from a smaller
number in the samecolumn?
41. Describe two difficulties that youngstersencounterwhen
learning to add,subtract,multiply,and divide using HinduArabic numerals.Baseyour answeron difficulties that are
encounteredwhen performingthesecomputationsin bases
other than ten.
CriticalThi;kinsExercise
31,"".n)tDE;;.
42.Divide:
11
"
""IOUr
7
4
35. 2rou.)ioq*
ZIrou,
X 1.21es1
X 231ss.
largest
le next
I
23n'.
- L4au"
563r"u"n
- 164r"r"n
543$r"n
0
0
19.
2\ro*
0
0
,,
18.
29.
4
0
323g""
+ 421fi,e
475ewt
- 267.rn,
ln Exercises25-34, muhiply in the iadicated base.
25.
n.
26. 34n
25,t*
"
X 4'l*
X 3nt"
3
0
a
L7.
1000t*o
101Mo
t
0
32*n"n
- 16r"""n
-
I
11,*o
+ LLtwo
16
22.
0
:
Exercises
Tachnology
In Exercises4346, usea scientific calculator that handlesdifferent baseconversionsto perform the calculation in the indicated
base.Note: You need to put the calculator in the appropriate
mode before you perform thesecalculations.If you are unsure
how to do this,consultyour owner'smarutal.
44.7I2eisht- 455"g11
g
43. 7l2"irtn1* 455"1rt
45. 65shs"nX 3ri^t""n
* 6si*t""n
46. 516.i*1""o
a*
^i
f.l
f
b oeJ*d]""1 s{#! €J-fi1t "a
-*_-'#*'---*''.@
2OO NumberTheoryrndthc RealNumbersystem. Chapter5
Solveproblemsusingthe
leastcommonmultiple.
tid.:
{$
A movie theater runs its films continuously.one movie runs for g0 minutes
andx
secondruns for 120minutes Both moviesbegin at 4:00p.u.when will the
movies
beginagainat the sametime?
{t
gr,
tolufloN
The shortermovie lasts80 minutegor 1.hour,20 minutes.It begins
at
4:00,so it will be shownagainat 5:20.The longermovielasts120minutes,or 2
hours.
It beginsat 4:00,so it will be shownagainat 6:00.we are askedto find when
the
movies wiu begtn again at the same time. Therefore,we are looking for the
least
commonmultiple of 80 and 120.End the leastcommonmultiple and then add
this
numberof minutesto 4:00p.u.
Beginwith the prime factorizationsof 80 and 120:
A
a.
80=24x5
I20:23x3x5.
Now select each prime factor, with the greatestexponent from each factorization.
=24 x 3 x 5 = 16 x 3 x 5 =240
Leastcommonmultiple
Thereforg it will take 240minuteg or 4 hours,for the moviesto begin againat the
sametime.By adding4 hours to 4:00r.v., they will start togetheragainar g:00p.rnr.
STUDYTIP
Example6 canalsobe solvedby makinga partiallist of startingtimesfor eachmovie.
ShorterMovie(Runst hour,20minutes):
t
4:00, 520, 6:40, 8:00,...
Iop.gelMovie(Runs2 hours):
:,-,1.,.:ftl.:,,,.1.r; ,t,, . , , 4:00, 6:00, 8:00,...
":-:,,,.
Thelist reveals
thatbothmoviesstarttogetheragainat g:00r.u.
A movie theater runs two documentaryfilms continuously.One documentaryrunsfor 40 minutesand a seconddocumentaryrunsfor 60minutes.Both moviesbeginat 3:00p.rnl.
Whenwill the moviesbesin asainat
the sametime?
&
Exercise
Set5.1
PralticeEiercises
14. 4h5,984
Useruleso/itthQility to determinewhethereachnumbergiven
in Exercb(s1-10$ divisibleby
\--l
a.2
b.3
c.4
d.5
e.6
f.8
h.l0
i.12.
9.9
t. 6944
2. 7245
3.21,408
4.25,025
5. 26,428 6. 89,001
7. 374.832 8. 347.712
17. 61104,538
20. 8128,096
23. 121s17,872
ls.'s138,814
18.61163,944
2r.91L7,378
24.12178s,172
16.5148,659
19.8120,104
n.9123,772
.^
In Exercise\!1-24,)se a calculatorto determinewhethereach
statementis tffi|alse.
If the statementis true,explain why this
is so usingone of the rulesof divbibility in Table5.1.
In Exercifts2541sfnd theprimefactorizationof eachcomposite
numben\-/
25.'15
26.45
n. 56
28.48
29.105
30. 180
31. 500
3L 360
33. 663
34. 510
35. 885
36.999
37.1440
38. 1280
39. 1996
40.1,575
il. 3159s8
41 3675
e. 6,126.120,
re_
\e4r,22r
12. 318142
13. 4110,612
42.831.6
43. 85.800
44. 30.600
5{
!r
Section
5.1 r Number
Theory:
Prime
andComposite
Numbers901
,"^\
45-Wd
h fuercis€
$.42and3f,
utesanda
he movies
, beginsat
Jr 2 hourl
whenthe
r the least
:n addrhis
{& 66and90
il.'12afi120
54.224and430
thegrmtestcommondivisorof thenutnben
46. ?5and70
4il. l6and42
49. 60and108
52.220and4N
55.240and285
5S,96and2l2
53. 342and380
56. 150and480
n nxercisbp
SZ-Afind theleastcommonmultipleof thenwnbers.
5E.25and70
52. nZanOS#
59.L6and42
6L 60and108
60. 66and90
62.96and2l2
63.72and120
64.220and2100 65.342and380
6.224 and430
67.240and285
68. 150and480
Exercises
Npplication
69. In Carl Sagan'snovel Contact,Ellie Arroway, the book's
heroine,hasbeenworking at SETI, the Searchfor ExtraterrestrialIntelligencg listeningto the crackleof the cosmos
Onenight,asthe radio telescopesare turned toward Vega,
they suddenly pick up strange pulses through the backeround noise. Two pulses are followed by a pause, then
ihree pulses,livg seven,
continuing through 97.Then it starts all over again. Ellie is
convinced that only intelligent life could generatethe structure in the sequenceof pulse$ "It's hard to imagine some
radiating plasmasending out a regular set of mathematical
signalslike this" What is it about the structure of the pulses
that the bookt heroine recognizesas the sign of intelligent
life? Asked in another way, what is significant about the
number ofpulses?
70. There are two speciesof inseclS Magicicada septendecim
and Magicacada tredecim, that live in the same environment.They have a life cycle of exactly 17 and 13 yearg
respectively.For all but their last year, they remain in the
ground feeding on the sap of tree roots Then, in their last
year, they emerge en masse from the ground as fully
formedcricketlike insectgtaking over the forest in a single
night.They chirp loudly, mate, eat, lay eggs,then die six
weekslater.
(Source:Marcusdu Sautoy,The Music of the Primes,Harper
Collins,2003)
bill (frve-dollaror ten-dollar).Whatis the largestnumberof
bills that you can placein eachstack?
74. Harley collects sports cards He has 360 football cards and
432 baseball cards. Harley plans to arrange his cards in
stacks so that each stack has the same number of cards
Also, each stack must have the same type of card (football
or baseball).Every card in Harley's collection is to be
ptacedin one of the stacksWhat is the largestnumber of
cards that can be placed in each stack?
75. You and your brother both work the 4:00 p.M.to midnight
shift.You haveeverysixth night oftYour brother hasevery
tenth night off. Both of you were off on June1.Your brother would like to seea movie with you.When will the two of
you havethe samenight offagain?
.
76. A mo'vietheater runs its films continuously.One movie is a
'Short documentary that runs for '40 minutes The other
, movie is a full-length feature that runs for 1@minutes Each
film is shown in a separatetheater.Both moviesbegin at
noon.When will the moviesbegin again at the sametime?
77. Two people are jogging around a circular track in the same
direction. One person can run completely around the track
in L5 minutesThe secondpersontakes 18 minutes If they
both start running in the sameplace at the sametime, how
long will i1 take them to be logetheratthis plrye if theyqontinue to run?
78. Two people are in a bicycle race around a circular track.
One rider can race completely around the track in 40 seconds.The other rider takes 45 seconds.If they both begin
the raceat a designatedstarting point, how long will it take
them to be togetherat this starting point againif they con.
tinue to race around the track?
torization.
;ain at the
rt 8:00p.pr.
;r
I
I
II
_j
Jne docuor 60 minin againat
q
ffi
148,6-5e
lzoJoq
l|23,772
h composite
3
60
99
)i)
0,600
a. Supposethat the two specieshave life cyclesthat are not
prime, say 18 and 12 years, respectively.List the set of
multiplesof 18that are lessthan or equalto 216.List the
set of multiplesof 12 that are lessthan or equal to 2L6.
Over a 2l6-year period how"nqny. times will the two
speciesemergein the sameyear and competeto share
the forest?
b. Recall that both specieshave evolved prime-number
life cycles,17 and L3 years,respectively.Find the least
commonmultiple of 17 and 13.How often will the two
specieshaveto sharethe forest?
c' Compareyour answersto parts(a) and (b).What explanation'can yog qffgr for each ppeciep.
having a prime
iiumber of yearsai the length of its life cycle?
71. A relief worker needsto divide 300bottlesof water and 144
cansof food into groupsthat eachcontainthe samenumber
of items.Also, eachgroup must havethe sametype of item
(bottledwater or cannedfood). What is the largestnumber
of relief suppliesthat can be put in eachgroup?
72. A choral director needsto divide 180men and 144women
into all-male and all-female singing groups so that each
group has the samenumber of people.What is the largest
numberof peoplethat can be placedin eachsinginggroup?
73. You have in front of you 310 five-dollar bills and 460 tendollar bills.Your problem:Placethe five-dollarbills and the
ten-dollar bills in stacksso that each stack has the same
number of bills, and each stack containsonly one kind of
!(riting in Mathematics
79. If a is a factor of c, what doesthis mean?
E0. Why is 45 divisibleby 5?
81. What does"c is divisibleby b" mean?
82. Describe the difference between a prime number and a
compositenumber.
Seaion5.2 r The lntegers;Order of Opentions
tl1
Simplify: 6z - 24 + 22.3 + 1..
tOLWlO}l There are no groupingsymbols.Thus,
we begin by evaluatingexponential expressions.Then
we multiply or divide.Finally,we add or subtract.
67-24+22,3+L
:36-24+4.3+1
Evaluat e e><p
o nent ial et<preeai o ne:
62 = 6,6 = 36and22 = 2.2 = 4,
=36-6,3+L
Terform lhe multipliaaliona and diviaions
fuom lettto right Startwrth 24 + 4 = 6.
=36-18+1
:18+1
N o wd o t h e m u l t i ? l i a a l i o f t 6 . 3
= 16.
Finally,pertorm the additiona and Eufuraalione
rig,hL Oublraat: 36 - 16 = 18.
lrom leftto
=L9
Add;18I1=19,
&
Simplify: 72 - 48 + 42.5 + 2.
vn below
Simplify: (-6)2 - (s - 7)2(-3).
of opera.
lesstates
addition.
and 5 is
JOLUTPN Becausegrouping symbolsappear,we perform the operation within
pirenthesesfirst.
(-6)'-(s-7)'z(-3)
= (-6)2- ?2)r<-3)
hould be
Work inside parenLheseafir aN:
5-7-5+(*7)=-2.
= 36- 4(:3)
II
"
'.Evalaateev,?
c nsnLiale><preaeio
na:
:36 - (-t2)
:48
II
/orKlngI
I
(-G)'=(-d(-o1
=56
and(-2)2=(-2)(-2)=q.
Muliiply:
4(-3) = *12.
- (-12) = 36 * 12 = 48.
Subr.racN:36
&
Simplifyr('8)' _- (10- i3)'z(-2).
: OCCUr;.I
-J
rcy occur
Exercise
Set5.2
Practice
Exercises
In Ex"rcis{frtart
by drawinga numberline that showsintegersfrom \S.re4. Then graph eachof the
following integerson
your numberline
1.3
tq
In 6rc7gise€-d
between
the@s
s. -2e7
3. -4
4. -2
insert either < or > in the shaded area
rc makethestatementtrue.
6. -1ffi 13
7.-13s-2
9. 8 H 50
8. -1ffi -13
il. -100 ff 0 _r
12. 0 F.[-300
'-\
In Exercise{1j-18,flnd theabsolutevalue.
\Y
--
rr. l-141
16.116l ^
.a\
14.l-161
17.l-3oo,oool
rs.-7 + (>/
20.-3 + (-4)
10.7#-9
$. l14l
il. l-1,000,0001
In Exercis$ 19-30lfind eachsum.
?L 12+ (-8)
219 NumberTLeoryand
the RealNumbcrSystane Chapter5
2L t3 + (-s)
2s. -9 + (+4)
23.6 + (-9)
26.-7 + (+3)
24"3 + (-rL)
n. -9 + (-9)
In Exercises67-80,firu| eachquortent,or, if applicabk, $atethat
the expressionis undefined
2E.-r3 + (-13)
^
2e.e + (-9)
30.13+ (-13)
n.+
In Exercis$ 1aTfiyfo rm the indicated subt actiotL
31. 13 - 8\-/
32. L4 - 3
33. 8 - t5
67.+
68.+
7s.+
?6.0
n.+
7't. (-480)+ 24
7e.(46s)+ (-1s)
6e.+
n.+
70.*
74.+
x.3 - (-17)
3e.-12 - (-3)
q. -$ - (-2)
,nr
4r. -rr - r'1
42. -t9 - 21
Usethe olffiolfiperations
in Exerdses
\ / 81-98.-\
Er.7+ 6\-/
83.(-s) - 6(-3)
8e -5 + (-3).8
84.-8(-3) - 5(-6)
43.6(-e)v
4. s(-7)
45. (-7X-3)
8s.6-4(-3)-s
46.(-8X-5)
48. (-3X10)
86.3-7(-L)-6
8 8 . 3- 9 ( - 1 - 6 )
4e.(-13X-1)
5L 0(-5)
8 7 . 3 -s ( - 4 - 2 )
8 e .( 2 - 6 X - 3 - 5 )
er. 3(-42 - 4?3)'
n. -6 - (-17)
In Exercifts43-52)findeachproduct.
se 0(-8)
/-\\
to ftnd the value of each exprexion
e 0 .e - 5 ( 6 - 4 ) - 1 0
n. s?r2 * 2(z1z
e3.(2- q2 - Q-T2
e4.(4- 6)2- (s - ef
9s.6(3- s)3- 2(1- 3)3
In Exercis( 5346fvaluate eachexponentialexpression.
\{{
53. 52
s6. (-6)2
62
ss. (-5)2
57. 43
59.23
se.(-sf
60.(-4f
96.-3(-6 + 8), - 5(-3 + 5)3
61. (-5)4
62. (-q4
63. -34
64 -t'
6s. (-3)4
66. (-ly
n.82-16+22,4-3
9S.1d - 100+ s2.2- (-3)
ApplicationExercises
,
'
Temperaturessometimesfall belowzero.A combination of low temperatureand wind makesit feel colder than the acfital temperatureThe
table showshow cold it feels whenlow temperaturesare combined with dffirent wind.speeils
wtxDcHrLt
Wind
(mph)
35
30
25
20
Temperature (" F)
5
15 10
0
-5
-10
-15
-20
31
25
19
13
7
-1
10
)1
21
15
9
3
-4 -10 -16 -22 -28 -35 -4t
15
25
19
13
6
0
-2
5
n
24
17
11
25
23
16
9
4
-7
d
7& (-300)+ 1z
80.(-se4)+ (-18)
3s.4 - (-10)
38.-4 - (-1e)
34.9-20
T
d
-5 -11 -1.6 -22 -28 -34
-73 -79 -26 -32 -39 -45
-9 -15 -22 -29 -35 -42 -48
3 -4 -1r -r7 -24 -3r -37 -44 .-5I
Sorrcc National Weather Seryice
Use the information from the table to solve Exercises99-100.
99. Write a negativeinteger that indicateshow cold the temperaturefeelswhenthe temperatureis 15oFahrenheitand
the wind is blowing at 20 miles per hour.
lfi). Write a negativeintegerthat indicateshow cold the temperaturefeelswhen the temperatureis 10oFahrenheitand
the wind is blowing at 15miles per hour.
101. The greatesttemperaturevariation recordedin a day is
1@ degreesin Browning,Montana, on January23,7916.
The low temperature was -56oF. What was the high
temperature?
102. ln Spearfish,SouthDakota, on January22,7943,thetem'
perature rose 49 degreesin two minutes. If the initial
temperaturewas -4oF, what was the high temperature?
103. The peak of Mount Kilimanjaro, the highest point in
Africa, is 79321feet abovesealevel.Qattara Depression'
Egypt, the lowest point in Africa, is 436 feet belowsea
level.Whatis the differencein elevationbetweenthepeak
of Mount Kilimanjaro and the QattaraDepression?
104. The peak of Mount Whitney is 14,494feet abovesealevel
Mount Whitney can be seendirectly aboveDeath Vallel'
which is 282feet below sealevel.What is the differencen
elevationbetweenthesegeographiclocations?
I
a
J
Numbers225
Seaion5.3 t TheRational
We canrepeatthe procedureof Example13 and find a rational numberhalfway
betweenI and |. Repeatedapplicationof this procedureimplies the followingsurprising result
reratorand
ber aa an
denamina-
Between any two given rational numbers areinfinitely many rationalnumbers.
&
Find a rational number halfwaybetweenI ana ].
t and
Pul
)aat corn-
Number
ProblemSolvingwith Rational
Solveproblemsinvolving
rationalnumbers.
A common applicationof rational numbersinvolvespreparingfood for a different
number of servings than what the recipe gives The amount of each ingredient can
be found asfollows:
Amount of ingredient needed
desired serving size
= -.
x ingredientamountin the recipe.
#
reclpeservmgsrze
E
ffi
A chocolate-chipcookie recipefor five dozencookiesrequiresI cup sugar.If you
want to make eight dozencookies,how much sugaris needed?
Amount of sugarneeded
tO$nON
)n rational
.onalnum-
desiredservingsize
slze
reclpeservrng
8Aez6 -3C U D
54ffi.4
lnumber
sugaramountin recipe
X
The amount of sugar,in cups,neededis determined by multiplying the rational
numbers:
8 ..3
8 ' 3 - 2 4- 6 ' {
s ' q : s ' q =6 : i A
nbersis to
rl numbers
- . ,'Li '
=
Thus,1| cupsof sugaris needed.(Dependingonthe measuringcup you are using,
you mqyaeed to round the sugaramountto 1i cups.)
A chocolate-chipcookierecipefor five dozencookiesrequirestwo eggs'
lf you want to make sevendrrze.ncookies,exactly how many eggsare
n."a"al Now round vour andwerto'a realistic number that doesnot
&
involve fractionalpartsof an egg.
Exercise
Set5.3
PracticZErerisqs
In Exer&es1-12. tduce eachrational number to its lowestterms.
Lig Y.X
i4
s- ' 1 ?
380
6.33
LO.H
3.,,D
4.#
q
"
8'*@
.@108
lt*3
n.#
/-\
In Exercitcs 13-18*onvert eachmixed number to an imprcp'
erfractionY
'3-21
16.-62
14. La'l;
ft.
toa
"16
q3
15. ' )
18. I I G
226 . NumberTheoryand
the RcalNumberSyste6o Chapter5
In Exercis$ pQi,\onvert
number. \-/
te.?
n. -+
each improper fraction to a mixed
2L-ry
z4X
20.+
ts.H
ApplicationExerciscs
The circle graph shows the breakdown of the number of cotlntries in the world that are free, partly free, or not free Uss4r,
inforrnation in the graph to solve Exercises101-102.
/z-\
In Exerci^sS
25-j6)expresseachrationalnumberasa decimal.
2s.? H.i
2e.l
s'*l
A. 'z+
$.+
2 7 .*
? s.*
31' ,r2
3Ltr
x.i
3s.+
World's Coun&ies by Siatus of Freedom
In Exercised 3748,Jexpress each terminating decimal as a
quotient of in\tge<If possible,reduceto lowat terms
37. 0.3
3& 0.9
39. 0.4
40. 0.6
4L 0.39
42.059
43. 0.82
44 0.64
45. 0325
6.0.6?5
{1.0.5399
48.0.7006
Sozrce l:rry
Dm
_7
In Exercb(s49-56,\tpresseachrepeatingdecimalasa quolient
of integersl,lfygiile reduceto lowestterms
49. 0.7
50. 0.1
sl. 0.9
5L 0.3
s3. 0.3-6
,/-\
/\\
s4. 0.8I
ss. 0.257- s6, 0.529
1'
In Exelspes57-9J/ perform the indicated operations.,lfpossible,
reducetheMer
to itslowes rcrms.
s7.e.+ s s . ; . *
60.(-l)(;)
er pi)(rl)
eo.i*l
6r.o!+ rfi
z'-'t
! + Z'
13 13
e.(-*X*)
61.(-3X-i) 6a(-i)(-i)'
ra.(z!)(r|)
At -!, * ?u
n. t] + zs,
?3.;-:
u. fi+fr
tr' * - (-":)
to.l+f;
oz.*l- + fi
s 4 . # -*
ss.i-i
e0.zl - zt
8s.*-:
eL(i-i)+;
z' "<' 1 12 - l - t \
\
l2l
zt.f+|
s7.+-4
6s.; + 3
e a- # * i
tt.fr+ft
7 4 . #itt.i+i
so.I + rrz
$.i;-3
8 6 .i-'1
ss.31- zl
tz.(l+ i) - (i * i)
In Exercises93-98,find a rational number halfway betweenthe
two numbersin eachpair.
e3.fandl
e6.Jand!
ea.I anal
et. -land-l
es.I andJ
98. -4 and-l
Berman and Brue Mwphy, Approaching
sacy, 4rh Editior' hentie Hall, 2003
101. What fractional part of the world's countries is free? Reduce the answerto its lowest terms.
102. What fractional part of the world's countries is not free?
Reduce the answerto tis lowest terms
In most societics,women say they prefer to marrv men whoare
older than themselveqwhereasmen say they prefer womenwho
are younger. Evolutionary psychologists attribute thesepreferencesto female concern with a parner! maftrtal resourcesand
male concem with a partner's fertility (Source:Davie M. Busq
PsychologicalInquiry 6,1-i0).The graph showsthepreferred
agein a man in five selecteilcountries Usethe information in the
graph to solve Exercises 103-106. Express each answerw a
mixednumber.
Preferred Age in a Mate
6
u)
96
o2?
.r
4
9!
.? sE 2
;>'
a^
EU
A-r
I
f; -u"-i
E6
^o
!q
-?
-4
e.E -s
f-6
6fr
-8
Zambia Colombia
Poland
Italy
United
States
Country
Dffirent operations with the same rational numbers usually
result in different ansnters.Exercises99-100 illustate somecurious exceptiors.
99. Showthat f; + f; anOf; x f; give the sameanswer.
100. Showthat # * i3 and $ + ,rEgivethe sameansrver.
Soarce:CaroleWadeand CarolTavris,Psychology,6th
Edition.PrenrictHall
2000
1.03. What is the difference between the preferred age in a maie
for women in Italy and women in the United States?
Itli
lrl
ll
!li
ili
ti
rii
rl
r Chapter5
NumberTheoryanddreReal
NumberSystem
2U
STUDY
TIP
c. llte smallestnumber that will produce a perfect squarein the denominator
of
Ybucan ratiodalize the denomi.1)
natorof ji
-v8
by multiplying
by
{r,because\/e. \/t : \/G = 4.Wemultiply
- J 1,
- ' -choosin*
--"""6
?^o
E - J by
4y i L ar r.
V8
12
$. Ho*.""r. it takes
more
V8
workto simplifytheresult.
V8
\/, = -----12 --=
n\/,
n\/t
v8 v2 v1.6 4
r
Rationalizethe denominator:
25
a. ---7
5
c.- -,
v18
b.
v10
lrrationalNumberr
andOtherKindlof Roott
Irrational numbersappear in roots other than squareroots.The symbol Vsentsthe cuberoot of a number. For example,
repre-
.2 = 8 and {A :4because 4.4.4 :
flA : Zbecause2.2
$.
Although thesecuberoots arerational numbergmostcuberoots arenot.Forexample,
* 6.0092because(6.0092)3x 216.995,notexactly217.
There is no end to the kinds of roots for nunlbers.For example,V- r.presents the fourth root of a number.Thus, l/-At:3
because3.3.3.3 = gl.
Although the fourth root of 81 is rational, most fourth roots,fifth roots,andso
on
tend to be irrational.
{Zn
Exercise
Set5.4
PracticeExercises
v.-E
w2
Evaluate eachexpressionin Errrrirb
r. v5
s. \/A
s. \h6s
2. vR
6. \4oo
N. \/ns
3. vT
7. \/1n
4. V4s
s. \EA
L ErrrdSEE\,
a calculatorwitha squareroot keytofind
a decimaldffimation for eachsquareroot.Roundthenumber displayedto thenearesta.tenth,b.hundredth,
e thousandth.
rr. \/173
14.\/ns264
u. \6n6
rs. \/;
13.\/iJ6t
16.\/G
3a-g
4r.4\/13- 6\/R
qs.t/i + t/s
44.\fr + {i
4s.4\/, - s{i + a\/t
46.6\/i + a\/1- rc\/t 4i. \/t + t/n
4s.\/, + t/fi
4s.\/so - \&
so.\,43 - t/-zs
" st. 3Vl8 + 5\40
40. 8\6 + 11\6
42. 6V17 - 8V17
s2.4V12+ 2V7s
$.
In Exercis{I
fl. \/n Y:
\6
zr. \/zso n \/1n
rs. \60
8.7\/28
20.\/n
24.3\/s2
In Exerci(s 2|-STperform the indicatedoperation.Simplify the
answerwhd*posdble.
zs.\/i.\f;
26.\/1s.{,
2i. \/6.\/6
2s.\/t:\6
3r.Vr.\/%
2s.\/1.rt
32.\/i.\R
n. \/12.\/t
n. g
3s.J:9
Vz
36.vlo
t/z
\/7<
34.:+
v3
v9
tg. zt/t + atfi
1-
;Vrz
_
1-
)t/tt
1-)-
s4. V3ooss.3\6 + 2\/n - 2\/48
;
it/zt
s6.2\/n+3\6 -\/tn
tn E*urifiszh
rationalize
thedenondnotor
n.*
'r.#,
*.#i
60.+
n+
e.JL
Vso
vs
v30
ur.+o 64.
iio u'\E ". r,E
the RcalNumberSystemo Chaptcr5
242 NumberThcoryand
ErerciseSet5.5
In Errrcis@lqt
narural\uxilrs
r.
all numbersfrom the givm setthat are
c. integers
b. whole numbers
d. rationalnumbers e. irrationalnumbers L realnumberc
tA,vz,t/too}
r. {-s,-!, o,o.zs,
z. {-1,-0.6,0,r,6,16}
{s,n,l7\
r. {-rr,-f,o,o.zs,
a.{-s,-0.3,0,lr,{o\
an exampleof a whole number that is not a natural
G)Ciu"
Vnumber.
CO Cin" an exampleof an integer that is not a whole number.
\.1
Ci". an exampleof a rational number that is not an integer'
@
($U Cin" an exampleof a rational number that is not a natural
v
number.
an integer,a whole
@Ciu. an exampleof a number that is
v number.and a natural number.
Cin. an exampteof a number that is a rational number,an
integer,anda real number.
fft)
v
number
r.'11f.)Give an exampleof a number that is an irrational
and a real number.
\-/
@)
V
Clue an exampleof a number that is a real numberrbutnot
^n irrational number.
Complete each statement in Exercb{{lfto
commutativeproPertY.
illustrate the
13.3+(4+5):3+(5+-)
M. \/t.4:4._
l s . e . ( 6+ 2 ) = e . ( 2 +_ )
i, nrrrr*rGnlo
eachstatement
Complqte
illustratethe
property.
associative
16.(3+7)+9:-+(7+-)
(s'_)
17. (1.s) . 3 = _.
Complete each statement in Errrrir@
\-/
distributiveproperty.
b illustrate the
1 8 .3 ' ( 6 + 4 ) = 3 ' 6 + 3 ' _
1 9 ._ ' ( 4 + 5 ) : 7 ' 4 + 7 ' 5
2 0 .2 . ( _ + 3 ) : 2 . 7 * 2 ' t
Use the disyib*ive:lt roperty to simpffi the radical expressions
in Exercis{?1-28r1
n. s(e+ ../i)
zt. l1(z * lr)
zs.r6(s + .'6)
zt. rfo(t/i + {o)
the nameof theproperty illustated.
In
PracticeExErcises
n. +(t + t/i)
u. t/o(t + t/3)
zo.t/i(t + r/1)
2s.\/10(\/, * Vlo)
29.6+(
(-4) + 6
30.11.(7 + 4) : 11.7 + L1.4
3L6+(2+7):(6+2)+1
n 6.(2.3):6.{3.2)
n. Q +3) + (4+ 5) = (4+5) + (2 + 3)
34.7' (n '8) = (11'8)'7
3s.2(-B + 6) = -16 a t,
36.-s(3 + 11)= -24 + (-88)
n. (n/l).{s = zf/1.{3)
3s.t/in = o{i
In Exercises3943,use two numbersto show that
39. the natural numbersarenot closedwith resPectto subtracion.
40. the natural numbers are not closedwith respectto division.
41. the integen are not closedwith respectto division.
42. the irrational numbers are not closed with respectto
subtraction.
43, the irrational numbers ar€ not closed with respectto
multiplication.
ApplicationExercises
44. Are first putting on your left shoeand then Putting onyour
right shoecommutative?
45. Are first getting undressed and then taking a shower
commutative?
z16.Give an example'of two things that you do ihat arenoi
commutative.
4?. Give an example of two things that you do that are
commutative.
48. Closureillustratesthat a characteristicof a set is not necessarilya characteristicof all of its subsetsThe realnumbers are closed with respect to multiplication, but the
irrational numberq a subsetof the real numbers,arenol'
Give an exampleof a set that is not mathematicalthathas
a particular characteristic,but which has a subsetwithout
this characteristic.
!(riting in Mathematics
49. What doesit meanwhen we say that the rationalnumben
are a subsetof the real numbers?
a
50. What does it mean if we say that a set is closedunder
given operation?
an
51. State the commutativeproperty of addition and gile
example.
give
52. State the commutativeproperty of multiplicationand
an example.
53. Statethe associativeproperty of addition andgiveanexample'
give
54. State the associativeproperty of multiplication and
an example.
and Scientific
Seaion5.6 . Exponents
Notation 951
ting large
'wed 96.8
)nal debt
BONUS
BLITZER
LastTheorem
Famat's
pierrede Fermat (1601-1665)was a lawyer who enjoyed studying mathematics In a margin
oioo.otlisbookqheclaimedthatnonaturalnumberssatisfy
an*V=cn
if n is geater than2.
lf n = 2, we ealnfind numbers satisfying the equation. For example,
gz*42:52.
/
satisfy
daimedthatnoheturalnumbers
However,Fery'rat
:.l+F=C,aa+ba=ca,
|,,'
t',
I
andsoon. i
Fermat claimed to have a:proof of his conjecture,but added, "The margin of my book
is too narrow to wite it down."S-omebelieve that he never had a proof and his intent was
to frustrate his colleagues
In June 1993,zl0-year-oldPrinceton math professorAndrew Wiles claimed that hq discovereda proof of the ttreorem.Subsequentstudy revealed flawg but Wiles corrected them.
In 1995,his final proof served as a classicexample of how great mathematiciansaccomplish
(AndrewWiles1951-)
greatthings:a combination ofgenius, hard work" frustration, and trial and error.
he mean-
ollars.At
llion), or
ul in the
2 dollars,
Set5.6
Exercise
PracticeExercises
r
,--\
to simplifueach
ln Exuc:u$1-12]tuepropertiesof expondnts
exprasion.Wexpress the answerin exponential
foryn Then
evaluaktheexprasian.
3.4-42
4 5.52
t 22-23
2.33.32
(zt)'
(10)t
8. (1')7
?.
6.
5.
Q\t
47
^t
f
18
e'
t''
to'
,^
*
;
"' ,o
.-F\
ln Exercis{I|24)usethezeroandnegative
exponent
rulesto
smptryeacftw(ession.
1r.30
ls. (-3)0
16. (-9)0
14. 90
-90
17.-30
20.3-2
tg. z-2
$.
23.2-s
24.2-6
22.2'3
,-\
In txucNeQ5-3L)ue propertiu of exponents
to simplifi each
expression.
Nuprest theanswerin exponential
form. Then
evaluate
theexpression.
zs.34.3-2
27. 3-t.3
26.zs.24
u.z-3.2 'n.i
,.#
u!
".;+#
9:6:f
"s0.
".
,t.
31.2.7x 102
32.4.7 x r03
33. 9.12x 105
/^)
':r?X:;ffif,if;f,,1:fi:X|y;Hi;:#i;:,H!:;:
3?.1 1 16s
40.8.6x 10-1
43.7.86x 10<
'16.s.84x 10-7
38. 1 x 108
4t z.ts x tol
44.4.63x to-s
39.7.gx to-t
42.3.t4 x ro-2
45. 3.18x 10-5
\...--/
34.
8.14
x 104 3s.8x 107
&
r.i
ln E*rrri"rGtrDress
the
r c number
n u r n D e rin
. n decimat
u e c u n u tnotation.
norat.ot,.
.
--
ream.At
illion, or
did each
a\
operation
and etpress
In Exercisfufrfil, performtheindicated
eachanswerin decimalnotation
6s. (5 x 102)(4x 104)
6j. (zx 103)(3x 102)
70.(4 x 108)(2x 10-"4)
69.(zx 10e)(3x 10-5)
7L. (4.t x 1d)(3 x 10-4) 72.(t.2 x 103)(2x 10-5)
x 10:
L2 x Lo:
'74.20
'B.
'-'
x 101s
x
10
4 LOz
x 102
x
18
15
104
__
-.
/o'
/5.
,.
10=,
9
5t]t
21.4-'"
I govern-
noution.
thenumberin scientific
In ExerciseS,(!fulxpress
50'
2700
49'
48' 530
3600
47' 370
53. 220,000,000
52. 64,000
5L 32,000
56. 0.014
55. 0.027
54. 370,000,000,000
59.0.00000293
58.0.00083
52.0.003?
62"630x 108
60. 0.000000647 6L 820x 105
65.2100x 10-e
64. 0.57x 10e
63. 0.41x 106
66. 97,ooox lo-u
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83. (0.0005x6,000,000) 84. (0.000015x0.004)
960
r Chapter5
NumberTheoryandtheRcalNumberSystcm
Exercise
Set5.7
In ExercisS 1-20,)write the first six terms of the arithrnetic
'rst term, a1, and common
ilffirencq il.
g
,
f
l
= 2
l. a1:
2. a1=J,/=3
3. a1= l@,1 = 29
5.a1:-7,d=4
7. a1: *400,d = 300
9.a1=7,d=-3
= -60
4.a1=300,d=50
6.a1=-$,f,=5
E, a1 = -JQS,af : 466
10. at = 9,d = -5
tl. at:2ffi,d
1 2 . a 1= 3 C I , d = - 9 0
1 3 .a 1= ] , a = l
14.a1-i,a=i
1 6a. r = i , a : - i
17. a1= 4.25,d: 0.3
18. at = 6.3,d= 0.25
19. a, : 4.5,d = -0.75
20. a, = 3.5,d = -1.75
1 5c. ,: i,a =i
In ExercisS 2140,Iind the indicated term for the arithmaic
sequencewi'llffienn,
a1, and commondifference,d.
21. Find a6,when a1 = 13,d : 4.
22. Finda16,whenq = 9,d = 2.
23. Find a5s,when a1 = l, / = 5.
?4. Frnda6o,rvhena1 = 8,d = 6.
25. End 4e,whenar = -5,d = 9.
5 9 .a 1= - 2 , r : - 3
6L a1=-6,r = -5
63a
. 1 = 1 , ,= 2
65a
. 1= i , , = i
fl. a, = -*,, = -q
69.at = 2,r = 0.1
54.a1= 2,r : -3
5 6 .a t : 2 0 , r : - 4
58. a1= 3000,r= -t
60.ar - -4,r = -2
62.ar=-8,r=-5
6 4 .a , = 1 , , = 2
6 6a. 1 = i , =
,i
-tr,,
=
= -Z
68.a1
70.a1=-1000,r=0.1
In ExerciQTl-\find
the indicated term for the geometic
sequencewitf;fitfi term, ar, and common ratio, r.
71. Find a7,when at = 4, r = 2.
72. Find as,when ar: 4,r : 3.
7 3 . F i n d a 2 s . w h e n a=t 2 , r : 3 .
74. Find a26,whena1 = 2,r :2.
75. Find a1ss.whenal = 50, r = 1.
76. Find a2oo,
whena1 = @.7 = 1.
26. Finddle,whenar= -8,d = 70.
27. Finda26,whenal= -40,d:5.
28. End a1e, whar'a1 : -60, d : 5.
29. Finddro,when
q= -8,d = 10.
53. ar = 3,r -- -2
55. cr : tQ,r = -4
{ 1 . a 1 = 2 0 0 0 , r =- l
77. find a7.whena.r = 5,r - -2.
,
'
l},d : -6.
31. find a6o,whena1: lJ, / = -3.
32. Finda7s,whenar = -32,d = 4.
33. Find412,whenar = 12,d = -5.
3 4 . F i n d c 2 s , w h eant = - 2 0 , d : - 4 .
30. Findall, whenar:
35; Find deo,when at = -70,d : -2.
36. Find a8o,when ar = I06,d = -72.
3 7 . F i n d a l 2 , w h ean1 = 6 , d = ) .
38. Find414,whena1 = g, i = L.
39. Finda56,whena1 = 1,4,d= -0.25.
40. Findallo,whenat= -12,d = -0.5.
---.-'
In Exercis$ 4148,)urite a formula for the generalterm (the nth
term) of eadv,itllinetic sequence.Then
usetheformula for a;to
find a2s,the20th term of the sequence.
7 8 . F i n d a a , w h e n c:l4 , r :
-3.
79. Find a3e,whena1 = 2. r = -1.
E0.Findaan,whenct= 6,r = -1.
81. Iinda6,whenar: -2,r = -3.
82. Finda5,*heno, = -5,r = -2.
83, Find a6,whena1 = 9,7 = L,
8 4 , F i n d a s , w h e n a=l l 2 , r : 1 .
85. Find c6,whena1 : lg,7 : *|.
86, Findaa,whena1 = ),7 = -L.
87, Finda4o,whenal= lffi,y = -1.
88. Find a3s,whenar = 8000,, : -).
89. Find c8, whena1 = 1,000,000,
r = 0.1.
90. Find as, whena1 = 40,000,r = 0.1.
,/-\t-\
4 1 .1 , 5 , 91, 3 , . . .
43.7,3, -1, -5, . . .
4 5 .a r = 9 , d = 2
47. at = -20,t| = -4
&.2,7,12,17,...
4 4 .6 , 1 . , - 4 , - 9 , . . .
46.ar=6,d=3
48.at = -70,d = -5
In Exercise\49-70,)orile the first six terms of the geometic
sequencewitMflfst
ternt,a1,and commonratio,r.
49.ar=4,r=2
51.a1=1000,r=1
5 0 .a r = 2 , r : 3
52.at=5000,r=1
In Exerci(s 91-98,lrite a formula for thegeneralterm Ahenth
term) of edeh,4w{etric sequence.Then usetheformula for a,to
find a7,theseventhterm of thesequence.
91.3,12,48,192,...
92.3,1,5,',|5,375,
...
e3.18,6,2,1,....
e4.12,$,tr,"...
o(
e6.5,-1,i,-*,
1s-?A-lt
-0.004,0.04,
-0.4,...
97. 0.0004,
-0.007,0.07,
-0.7....
98.0.0007.
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