D ttusrRATroN z
A geometric sequenceis also called a geometric progression. The constantmultiplier in a geometricsequenceis called the common ratio and is denotedby r. We may computer by dividing any term after the first by the precedingone.
ItruSTRATtON 6 1,2 ,4,8 ,16,32, 64,r 28 we have? : 2, t : 2, 2 : 2, E : 2, # : 2, f, = 2, andff : 2. The commonratior:2,
As with an arithmetic sequence,the numberof elementsin a geometrio sequenceis denotedby N, the first elernentis denotedby at, and the last elementis denoted-bydN.In the sequenceof Illustration 6, N : 8, a1 : I, andaa: 128. A geometricsequencecan be definedby giving the valuesof ar and N and a recursiveformula An+l:
Anl
from whibh every elementafter the first can be obtained from the preceding one.
D ttusrRATroNz Consider the geornetric sequence for which ar : I28, N : 5, and an+t : a,(-t). Then
ar: 128(-I) a, : -32(-I) -8 :-32 Therefore, the sequenceis
L28,-32,8, -2, i
ao: 8(-i)
- _2
a s: - 2 ( - i )
=t
762
CHAPTER12 TOPICSIN ATGEBRA
The generalgeometricsequence,for which the first elementis cr, the common ratio is r, and the number of elementsis N, can be obtainedby applyingthe recursiveformula. Startingwith the elementar, we obtain each successiveelementby multip$ing the precedingone by r. Doing this, we have a1, a1f, a1r2, a1r3, a1f4, . , . , aN
In the first five elementswe observethat eachelementis the product of ar and a power of r, wherethe exponentof r is one lessthan the numberof the element. Therefore our intuition suggeststhat the Nth (last) element is Qu :
QtfN-r,
The Nth elementof a geometricsequenceis given by aw:
arrN-l
The proof of this theoremis by mathematicalinductionand is left as an exercise.SeeExercise52.
Sequenceos q MathematicolModel A city has a current populationof 100,000.If the populationis sxpectedto increase10 percentevery five years,what is the expectedpopulation forty r' yearsfrom now? Solution
The population at the end of five yearsis expectedto be
: (1.10X100,000) 100,000+ 0.10(100,000) The expectedpopulation at the end of each successivefive-yearperiod is 1.10 times the population at the end of the preceding five-yearperiod. Hence we havethe geometricsequenceof nine elements (1.10)(100,000), (1.10)' (100,000), . . ., Qs 100,000, whereagis the expectedpopulationat the end offorty years.From Theorem 3 with N : 9, ar: 100,000,and r : 1.10,' as:
'
a- I
atf-
= 100,000(1.10)8 : (2.r44)L0s Conclusion: Forty yearsfrom now, the populationis expectedto be 214.400to thenearesthundred.
I2.6
ARITHMETIC AND GEOMETRIC SEQUENCES AND SERIES
763
P IUUSTRATIoN8 The sequence 2 , 6 , L 8 , 5 4 ,1 6 2 is a geometricsequencewith r : 3. From the deflnition the numbers6, 18, and 54 form a set of three geometricmeansbetween2 and 162. Becausethe sequence 2, -6, r9, -54,162 is alsoa geometricsequence(r = -3), the numbers-6, 18, and -54 form anothersetofthreegeometricmeanSbetween2and'I62.< If m is a geometricmean betweentwo numbersx and H then Xrffir!
l
is a geometricsequence.Therefore * _ ! x m tnz : xy This equationimplies that either both.r and y are positive or both tgnd y are negative.Furthermore,the equationhas two solutions:m = Y xy and m = -Y xy. Becausewe want the geometricmeanto be betweenthe numbers.r and y, we choosefor the value of ln the numberhavingthe samesign as x and y. Thereforewe havethe following definition.
764
CHAPIER12 TOPICSIN ATGEBRA
Find the geometric mean between each set of numbers: (a) 4 and 9; (b) - ro1and - i. Solution In eachpart let mbe Ihe geometricmean.We computem by applyingthe definition.
(b)m: -\4rE)(_F : -\4
(a)m : \/4 4 : .// 36 -6
:-Yi
: *+
By a generalizationof the definitionof the geometricmeanbetweentwo numbers,wedefinethegeometricmeanof asetof numbersxl,x2,x3, . . , xntobe the numberYxr.r-t . . . -".
P ru.usrRATroN 9 The geometricmean of the numbers4, 10, and 25 is
t44)1to)(z'= Vlooo =10 With any geometricsequence,there is an associatedgeometric series, which is the indicatedsum of the elementsof the geometricsequence.
D ru.usrRATroN ro
I
The geometricsequenceof Illustration 7 is
128,-32, 8, -2, i Associatedwith this sequenceis the geometricseries
128-32+8-2++ which can be written with siemanotationas 5
)i = L tzal-1;'-' Let Sr,'be the sum of N terrns of a geometricseries.Then S,,r: 4r * alr * af2 + a1r3+ . .. * afN-2 4 orrN-l
12.6 ARITHMETIC AND GEOTT,IEIRIC SEQUENCES AND SERIES
765
If we multiply both sidesof this equationby r, we have tSx :
* a1r2 * a1r3 I
af
arra + . . . + errN-l *
atrN
The sum of the first equationand - 1 times the secondgives Sl,t-f,$,,:At-AlfN
(1-r)Srv:Qr-arrN If 1 - r * O,wecandivideeachsideofthis equationbyI - rand obtain Qt
^
S":*
-
atfN
ifr+l
t - r
a{l-r\ ^ ^Srv=-# r - r
ifr+I
A formula for S,,,'in terms of ai, r, anda,.uis found by expressingarrN as r(a1rN-l). Doing this, we have ar -
^
r(arrN-t\
iir z I
l - r
From Theorem3,alrN-t = a". Thus sr-at--rau l - r
ifr*l
We haveproved the following theorem.
If a1,a2,a3,, . ., a,,s is a geometficsequence with commonratio r, and Sy=arlaz*az*...lap then a(I - r\ (i).Siv: L - r
ifr*I
and (ii) ,s" ' :
at.- fau l - r
if r # |
Find the sum of the geometric series : t r/!\i-r H
L\3/
CHAPTER12 TOPICSIN ATGEBRA
For the g i v e n s e r i e sa, r : 2 , Solution from Theorem4(i) -
J5
:
-
a(I
----;-
r:
j, and N:
5. Thus,
rs)
t - r
_ 2t1- (+)'l 1-i _ 2 ( r :h ) : ? - l '
dl
: 23? In the next examplewe useTheorem3 of Section6.2, which statesthat if P dollars is investedat an interestrate of 1001percent compoundedn times per year and if A" dollars is the amount of the investmentat the end of n interestperiods,then An:
r(t + t)"
as a Malhematical Model To create a sinking fund that will provide capital to purchasesome new equipment,a companydeposits$25,000into an accounton January1 every year for ten years. If the account earns 12 percent interest compounded annually, how much is in the sinking fund immediately after the tenth depositis made? Immediately after the tenth deposit is made, the tenth Solution paymenthas earnedno interest;the ninth paymenthas earnedinterestfor one year; the eighth paymenthas earnedinterestfor two years;and so on; and the first paymenthas earnedinterestfor nine years.To find the number of dollars in the fund immediately after the tenth payment,we apply the aboveformula for A, with P : 25,000,i : 0.1,2,and m: 1 to find the dollar amount from eachpayment.The results are as follows:
10thpayment: 25,000 payment: 12)l 25,000(1. 9th
(no interest)
(interestfor one yea4 n : I) Sth payment: 25,000(1.12)2 (interestfor two yearc;n : 2) lst payment:25,000(1.12)' (interestfor nine years;n : 9)
ItruSTRATtON 6 1,2 ,4,8 ,16,32, 64,r 28 we have? : 2, t : 2, 2 : 2, E : 2, # : 2, f, = 2, andff : 2. The commonratior:2,
As with an arithmetic sequence,the numberof elementsin a geometrio sequenceis denotedby N, the first elernentis denotedby at, and the last elementis denoted-bydN.In the sequenceof Illustration 6, N : 8, a1 : I, andaa: 128. A geometricsequencecan be definedby giving the valuesof ar and N and a recursiveformula An+l:
Anl
from whibh every elementafter the first can be obtained from the preceding one.
D ttusrRATroNz Consider the geornetric sequence for which ar : I28, N : 5, and an+t : a,(-t). Then
ar: 128(-I) a, : -32(-I) -8 :-32 Therefore, the sequenceis
L28,-32,8, -2, i
ao: 8(-i)
- _2
a s: - 2 ( - i )
=t
762
CHAPTER12 TOPICSIN ATGEBRA
The generalgeometricsequence,for which the first elementis cr, the common ratio is r, and the number of elementsis N, can be obtainedby applyingthe recursiveformula. Startingwith the elementar, we obtain each successiveelementby multip$ing the precedingone by r. Doing this, we have a1, a1f, a1r2, a1r3, a1f4, . , . , aN
In the first five elementswe observethat eachelementis the product of ar and a power of r, wherethe exponentof r is one lessthan the numberof the element. Therefore our intuition suggeststhat the Nth (last) element is Qu :
QtfN-r,
The Nth elementof a geometricsequenceis given by aw:
arrN-l
The proof of this theoremis by mathematicalinductionand is left as an exercise.SeeExercise52.
Sequenceos q MathematicolModel A city has a current populationof 100,000.If the populationis sxpectedto increase10 percentevery five years,what is the expectedpopulation forty r' yearsfrom now? Solution
The population at the end of five yearsis expectedto be
: (1.10X100,000) 100,000+ 0.10(100,000) The expectedpopulation at the end of each successivefive-yearperiod is 1.10 times the population at the end of the preceding five-yearperiod. Hence we havethe geometricsequenceof nine elements (1.10)(100,000), (1.10)' (100,000), . . ., Qs 100,000, whereagis the expectedpopulationat the end offorty years.From Theorem 3 with N : 9, ar: 100,000,and r : 1.10,' as:
'
a- I
atf-
= 100,000(1.10)8 : (2.r44)L0s Conclusion: Forty yearsfrom now, the populationis expectedto be 214.400to thenearesthundred.
I2.6
ARITHMETIC AND GEOMETRIC SEQUENCES AND SERIES
763
P IUUSTRATIoN8 The sequence 2 , 6 , L 8 , 5 4 ,1 6 2 is a geometricsequencewith r : 3. From the deflnition the numbers6, 18, and 54 form a set of three geometricmeansbetween2 and 162. Becausethe sequence 2, -6, r9, -54,162 is alsoa geometricsequence(r = -3), the numbers-6, 18, and -54 form anothersetofthreegeometricmeanSbetween2and'I62.< If m is a geometricmean betweentwo numbersx and H then Xrffir!
l
is a geometricsequence.Therefore * _ ! x m tnz : xy This equationimplies that either both.r and y are positive or both tgnd y are negative.Furthermore,the equationhas two solutions:m = Y xy and m = -Y xy. Becausewe want the geometricmeanto be betweenthe numbers.r and y, we choosefor the value of ln the numberhavingthe samesign as x and y. Thereforewe havethe following definition.
764
CHAPIER12 TOPICSIN ATGEBRA
Find the geometric mean between each set of numbers: (a) 4 and 9; (b) - ro1and - i. Solution In eachpart let mbe Ihe geometricmean.We computem by applyingthe definition.
(b)m: -\4rE)(_F : -\4
(a)m : \/4 4 : .// 36 -6
:-Yi
: *+
By a generalizationof the definitionof the geometricmeanbetweentwo numbers,wedefinethegeometricmeanof asetof numbersxl,x2,x3, . . , xntobe the numberYxr.r-t . . . -".
P ru.usrRATroN 9 The geometricmean of the numbers4, 10, and 25 is
t44)1to)(z'= Vlooo =10 With any geometricsequence,there is an associatedgeometric series, which is the indicatedsum of the elementsof the geometricsequence.
D ru.usrRATroN ro
I
The geometricsequenceof Illustration 7 is
128,-32, 8, -2, i Associatedwith this sequenceis the geometricseries
128-32+8-2++ which can be written with siemanotationas 5
)i = L tzal-1;'-' Let Sr,'be the sum of N terrns of a geometricseries.Then S,,r: 4r * alr * af2 + a1r3+ . .. * afN-2 4 orrN-l
12.6 ARITHMETIC AND GEOTT,IEIRIC SEQUENCES AND SERIES
765
If we multiply both sidesof this equationby r, we have tSx :
* a1r2 * a1r3 I
af
arra + . . . + errN-l *
atrN
The sum of the first equationand - 1 times the secondgives Sl,t-f,$,,:At-AlfN
(1-r)Srv:Qr-arrN If 1 - r * O,wecandivideeachsideofthis equationbyI - rand obtain Qt
^
S":*
-
atfN
ifr+l
t - r
a{l-r\ ^ ^Srv=-# r - r
ifr+I
A formula for S,,,'in terms of ai, r, anda,.uis found by expressingarrN as r(a1rN-l). Doing this, we have ar -
^
r(arrN-t\
iir z I
l - r
From Theorem3,alrN-t = a". Thus sr-at--rau l - r
ifr*l
We haveproved the following theorem.
If a1,a2,a3,, . ., a,,s is a geometficsequence with commonratio r, and Sy=arlaz*az*...lap then a(I - r\ (i).Siv: L - r
ifr*I
and (ii) ,s" ' :
at.- fau l - r
if r # |
Find the sum of the geometric series : t r/!\i-r H
L\3/
CHAPTER12 TOPICSIN ATGEBRA
For the g i v e n s e r i e sa, r : 2 , Solution from Theorem4(i) -
J5
:
-
a(I
----;-
r:
j, and N:
5. Thus,
rs)
t - r
_ 2t1- (+)'l 1-i _ 2 ( r :h ) : ? - l '
dl
: 23? In the next examplewe useTheorem3 of Section6.2, which statesthat if P dollars is investedat an interestrate of 1001percent compoundedn times per year and if A" dollars is the amount of the investmentat the end of n interestperiods,then An:
r(t + t)"
as a Malhematical Model To create a sinking fund that will provide capital to purchasesome new equipment,a companydeposits$25,000into an accounton January1 every year for ten years. If the account earns 12 percent interest compounded annually, how much is in the sinking fund immediately after the tenth depositis made? Immediately after the tenth deposit is made, the tenth Solution paymenthas earnedno interest;the ninth paymenthas earnedinterestfor one year; the eighth paymenthas earnedinterestfor two years;and so on; and the first paymenthas earnedinterestfor nine years.To find the number of dollars in the fund immediately after the tenth payment,we apply the aboveformula for A, with P : 25,000,i : 0.1,2,and m: 1 to find the dollar amount from eachpayment.The results are as follows:
10thpayment: 25,000 payment: 12)l 25,000(1. 9th
(no interest)
(interestfor one yea4 n : I) Sth payment: 25,000(1.12)2 (interestfor two yearc;n : 2) lst payment:25,000(1.12)' (interestfor nine years;n : 9)
