A black line master of Example 3 You Try is on provided on page 10 for duplication or use with a projection system.


 Lenard Watson
 2 years ago
 Views:
Transcription
1 Grde Level/Course: Algebr Lesso/Uit Pl Nme: Geometric Sequeces Rtiole/Lesso Abstrct: Wht mkes sequece geometric? This chrcteristic is ddressed i the defiitio of geometric sequece d will help derive the recursive formul. Studets will write the recursive d explicit formuls for geometric sequeces. Timefrme: clss periods Commo Core Stdrd F BF.: Write rithmetic d geometric sequeces both recursively d with explicit formul use them to model situtios d trslte betwee the two forms. Notes: The Wrm Up is o pge. A blck lie mster of Exmple You Try is o provided o pge 0 for duplictio or use with projectio system. There re two forms of the recursive formul forms re used iterchgebly i this lesso. r d r. These two Istructiol Resources/Mterils: Wrm Up Blck lie mster Exmple visul id Idex Crds (optiol) Pge of 0//
2 Lesso: Thik Pir Shre: Describe the ptter i ech sequece. )... b)... TPS Aswers: ) Ech term is times the previous term. (Also the sequece is ot rithmetic.) b) Ech term is more th the previous term. (Also the sequece is rithmetic.) REVIEW from the Arithmetic Sequece Lesso: A sequece is list or ordered rrgemet of umbers figures or objects. The members which re lso elemets re clled the terms of the sequece. A geerl sequece c be writte s... where is the first term is the secod term d so o. The th term is deoted s. A geometric sequece is list of umbers i which the rtio of y term to the previous term is costt. The costt rtio is clled the commo rtio is deoted by r. r Exmple : Determie if the sequece is geometric. Justify your swer. Use the defiitio d check if ll rtios re the sme. Sequece... Coclusio Sice ll the rtios re costt (costtly ) the sequece is geometric d the commo rtio is. Pge of 0//
3 Try... Sice ll the rtios re differet d NOT the sme costt the the sequece is NOT geometric. Aother wy to represet the rtio symboliclly is to use rrows showig the multiplictio from oe term to the ext Thik Pir Shre: Expli to your prter wht the equtio C we rewrite this equtio i other form? r is used for d how to use it? Derive the Recursive Formul of Geometric Sequece Solve r for : r r r r r r (or r ) re equivlet equtios. The equtio r is clled the recursive formul of geometric sequece. It defies the ext term s the previous term times the commo rtio. It c be used to geerte the terms of geometric sequece oe term t time. Pge of 0//
4 Exmple : Give the geometric sequece ) Fid the ext term. Sice we kow the sequece is geometric there is commo rtio. Wht is it? Use r : r. Usig 00 r r 0. 0 Method Sice we kow the fourth term d the commo rtio we c use the recursive formul to fid the fifth term. Use r. Substitute d simplify Recursive Formul: Use the equtio derived o the previous pge. The recursive formul is used to fid the ext term i the sequece by multiplyig the previous term by the commo rtio. r 0 The recursive formul is 0 *Remember there re two prts to this formul. Method Rewrite ech term i terms of the first term d the commo rtio ? Explicit Formul: Use the ptter to fid y term. The explicit formul is used to fid y term i the sequece without kowig the previous term *Use more terms if eeded to get studets to see this ptter. Thik Pir Shre: Which formul(s) c be used to fid 0? Which formul would be most efficiet? Justify your swer. Both formuls c be used to fid the 0 th term. Method is ot very time efficiet s you would eed to fid ll the terms ledig up to 0. Method is the most direct pproch sice you oly eed to kow the vlue of which i this cse is 0. b) Fid 0. If 0 the Therefore. Pge of 0//
5 Discuss: Should we evlute or leve it s is? Refer bck to the defiitio of geometric sequece d geerlize : A geometric sequece c be writte s r r r r r... r r... where is the first term or iitil coditio d r is the commo rtio. Like rithmetic sequeces geometric sequeces lso hve recursive d explicit formuls. The formuls for rithmetic sequeces re provided for review d pplictio. Type of Sequece Recursive Formul (or rule) Explicit Formul (or rule) Arithmetic + d + ( )d The commo differece is d. Geometric where is give r r The commo rtio is r. where is give Exmple : Write the recursive d explicit formuls for the sequece... Thik Pir: Wht kid of sequece is...? Which formuls do we use? The sequece is geometric so use the geometric formuls. Commo Rtio Recursive Formul (or rule) Explicit Formul (or rule) r Use r d replce the r vlue d stte the first term. Use r d replce the r vlue d the first term. r or where Pge of 0//
6 Exmple b (optiol): Use both formuls to fid. Use Recursive Formul (or rule) where d. Use Explicit Formul (or rule) where. Idetify the terms up to :?? st : Fid d : Fid 0 ( ) ( ) TRY (with Solutios): Decide whether the sequece is rithmetic geometric or either. Fid the ext term. The write the recursive d explicit formuls. Sequece Next term Commo Rtio or Differece Recursive Formul (or rule) Explicit Formul (or rule) Prter A Prter B Techer s Choice (Geometric) (Geometric) (Arithmetic) r ( ) where r ( ) d + where ( ) where ( ) ( )( ) + or 7 Pge of 0//
7 Exmple : Suppose you drop bll from height of 00 cm. It bouces bck to 80% of its previous height. ) About how high will the bll go fter its fifth bouce? Iitil height of bll: After first bouce: 80% of 00 cm After d bouce: 80% of 80 cm 00 cm 0.80(00 cm) 80 cm 0.80(80 cm) cm After rd bouce: 80% of cm 0.80( cm). cm After th bouce: 80% of. cm 0.80(. cm) 0.9 cm After th bouce: 80% of 0.9 cm 0.80(0.9 cm).78 cm Techer Note: Omit the uderlied vlues d prompt studets to idetify the correct vlue tht belogs i ech blk. Therefore the bll will reboud bout.8 cm fter the fifth bouce. Thik Pir: Wht kid of sequece is creted by these heights? How do you kow? Expli wht the 00 cm d the 80% represet. b) Write the recursive d explicit formuls for the geometric sequece geerted by these heights. To write both formuls idetify the commo rtio d the first term: r 0. 8 d 00 Recursive Formul: ( ) 0.8 where 00 Explicit Formul: ( ) TRY: Model the situtio below usig recursive d explicit formul. At the begiig of experimet there re 00 bcteri coloies. The umber of coloies doubles ech hour. *Note: This problem is from the wrm up. Recursive Formul: where 00 Explicit Formul: ( ) 00 Pge 7 of 0//
8 Thik Pir Shre/Discussio Questios: Why is it ecessry to idetify the first term i the recursive formuls? If the first term is ot idetified the the formul represets y sequece tht hs the sme commo rtio. For exmple ( ) represets the sequeces d How do geometric sequeces with positive commo rtio compre to geometric sequeces with egtive commo rtio? The terms of geometric sequece with positive commo rtio re the sme sig (ll positive or ll egtive) wheres the terms of geometric sequece with egtive commo rtio lterte sigs. How re geometric sequeces similr to expoetil fuctios? The explicit formuls of geometric sequeces re expoetil fuctios. How is geometric sequece differet from expoetil fuctio? A expoetil fuctio is cotiuous d the domi is ll rel umbers. A geometric sequece is collectio of poits tht re ot coected which mke it ot cotiuous d the domi is ll turl.... umbers { } Is geometric sequece fuctio? Yes geometric sequece is fuctio whose domi is ll turl umbers {... }. Therefore the explicit formul of geometric sequece c be writte i fuctio ottio. Arithmetic d Geometric Sequeces d Fuctio Nottio: Sequece Recursive Formul i Fuctio Nottio Explicit Formul i Fuctio Nottio Arithmetic f ( ) f ( ) + d with f ( ) give f ( ) f ( ) + ( )d Geometric f ( ) f ( ) r with f ( ) give ( ) ( ) f f r Pge 8 of 0//
9 Exit Ticket Optios: Extesio Activity/Optio #:. Ech studet cretes their ow geometric sequece d writes it o oe side of idex crd.. Ech studet writes the recursive d explicit formuls for their sequece o the other side of crd.. Studets write their mes o oe side of the idex crd d submit them to the techer.. Techer checks crds for ccurcy.. Techer returs crds to studets o differet dy so they c prticipte i the Quiz Quiz Trde Activity. Techer tells studets to std up d pir up. Prter A quizzes B. Prter B swers. Prter A prises or coches. Prters switch roles. Prters trde crds. After techer specified time repet the steps bove with ew prter. Optio #: Determie if ech sttemet is true or flse. ) The sequece is geometric. True Flse ) The sequece 8... is geometric. True Flse ) The recursive formul 80 represets the sequece True Flse ) The explicit formul ( ) represets the sequece... True Flse ) The recursive formul ( ) ( ) represet differet sequeces. d the explicit formul True Flse ANSWERS: F T T F F Pge 9 of 0//
10 Type of Sequece Recursive Formul (or rule) Explicit Formul (or rule) Arithmetic + d + ( )d Hve commo differece d. Geometric where is give r r Hve commo rtio r. where is give Sequece Next term Commo Rtio or Differece Recursive Formul (or rule) Explicit Formul (or rule) Circle Sequece Type: Geometric Arithmetic Neither... Circle Sequece Type: Geometric Arithmetic Neither... Circle Sequece Type: Geometric Arithmetic Neither Circle Sequece Type: Geometric Arithmetic Neither Pge 0 of 0//
11 Exmple Visuls Pge of 0//
12 Wrm Up CCSS: F LE. y CCSS: F BF. Use the grph below to determie if ech sttemet is true or flse. Give the rithmetic sequece 8...: A) f ( x) x + True Flse B) f ( x) + x True Flse ) Describe why the sequece is rithmetic. x C) f ( x) True Flse b) Idetify the commo differece. D) f ( x) x+ True Flse c) Write the recursive formul. E) f ( x) 0 True Flse d) Write the explicit formul. Fid the vlue of f ( ). CCSS: F LE. cotiued Curret: x y At the begiig of experimet there re 00 bcteri coloies. The umber of coloies doubles ech hour d is recorded. Complete the tble. Redig Number of Coloies x 00 0 Pge of 0//
13 Wrm Up Solutios CCSS: F LE. y CCSS: F BF. Use the grph below to determie if ech sttemet is true or flse. Give the rithmetic sequece 8...: A) f ( x) x + True Flse B) f ( x) + x x C) f ( x) D) f ( x) x+ E) f ( x) 0 f ( ) True True True True Flse Flse Flse Flse ) The sequece is rithmetic becuse there is commo differece betwee cosecutive terms. b) The commo differece is. c) where d) ( ) or + CCSS: F LE. cotiued Curret: x y At the begiig of experimet there re 00 bcteri coloies. The umber of coloies doubles ech hour d is recorded. Complete the tble. Redig Number of Coloies x Pge of 0//
A function f whose domain is the set of positive integers is called a sequence. The values
EQUENCE: A fuctio f whose domi is the set of positive itegers is clled sequece The vlues f ( ), f (), f (),, f (), re clled the terms of the sequece; f() is the first term, f() is the secod term, f() is
More informationArithmetic Sequences
Arithmetic equeces A simple wy to geerte sequece is to strt with umber, d dd to it fixed costt d, over d over gi. This type of sequece is clled rithmetic sequece. Defiitio: A rithmetic sequece is sequece
More informationCHAPTER 7 EXPONENTS and RADICALS
Mth 40 Bittiger 8 th Chpter 7 Pge 1 of 0 CHAPTER 7 EXPONENTS d RADICALS 7.1 RADICAL EXPRESSIONS d FUNCTIONS b mes b Exmple: Simplify. (1) 8 sice () 8 () 16 () 4 56 (4) 5 4 16 (5) 4 81 (6) 0.064 (7) 6 (8)
More informationRepeated multiplication is represented using exponential notation, for example:
Appedix A: The Lws of Expoets Expoets re shorthd ottio used to represet my fctors multiplied together All of the rules for mipultig expoets my be deduced from the lws of multiplictio d divisio tht you
More informationGaussian Elimination Autar Kaw
Gussi Elimitio Autr Kw After redig this chpter, you should be ble to:. solve set of simulteous lier equtios usig Nïve Guss elimitio,. ler the pitflls of the Nïve Guss elimitio method,. uderstd the effect
More informationShowing Recursive Sequences Converge
Showig Recursive Sequeces Coverge Itroductio My studets hve sked me bout how to prove tht recursively defied sequece coverges. Hopefully, fter redig these otes, you will be ble to tckle y such problem.
More informationThe Fundamental Theorems of Calculus
The Fudmetl Theorems of Clculus The Fudmetl Theorem of Clculus, Prt II Recll the Tkehome Messge we metioed erlier Exmple poits out tht eve though the defiite itegrl solves the re problem, we must still
More informationSummation Notation The sum of the first n terms of a sequence is represented by the summation notation i the index of summation
Lesso 0.: Sequeces d Summtio Nottio Def. of Sequece A ifiite sequece is fuctio whose domi is the set of positive rel itegers (turl umers). The fuctio vlues or terms of the sequece re represeted y, 2, 3,...,....
More informationChapter 04.05 System of Equations
hpter 04.05 System of Equtios After redig th chpter, you should be ble to:. setup simulteous lier equtios i mtrix form d vicevers,. uderstd the cocept of the iverse of mtrix, 3. kow the differece betwee
More information8.1 Arithmetic Sequences
MCR3U Uit 8: Sequeces & Series Page 1 of 1 8.1 Arithmetic Sequeces Defiitio: A sequece is a comma separated list of ordered terms that follow a patter. Examples: 1, 2, 3, 4, 5 : a sequece of the first
More information1 The Binomial Theorem: Another Approach
The Biomial Theorem: Aother Approach Pascal s Triagle I class (ad i our text we saw that, for iteger, the biomial theorem ca be stated (a + b = c a + c a b + c a b + + c ab + c b, where the coefficiets
More information1.2 Accumulation Functions: The Definite Integral as a Function
mth 3 more o the fudmetl theorem of clculus 23 2 Accumultio Fuctios: The Defiite Itegrl s Fuctio Whe we compute defiite itegrl b f (x) we get umber which we my iterpret s the et re betwee f d the xxis
More informationEXPONENTS AND RADICALS
Expoets d Rdicls MODULE  EXPONENTS AND RADICALS We hve lert bout ultiplictio of two or ore rel ubers i the erlier lesso. You c very esily write the followig, d Thik of the situtio whe is to be ultiplied
More informationMATHEMATICS FOR ENGINEERING BASIC ALGEBRA
MATHEMATICS FOR ENGINEERING BASIC ALGEBRA TUTORIAL  INDICES, LOGARITHMS AND FUNCTION This is the oe of series of bsic tutorils i mthemtics imed t begiers or yoe wtig to refresh themselves o fudmetls.
More informationFOURIER SERIES PART I: DEFINITIONS AND EXAMPLES. To a 2πperiodic function f(x) we will associate a trigonometric series. a n cos(nx) + b n sin(nx),
FOURIER SERIES PART I: DEFINITIONS AND EXAMPLES To periodic fuctio f() we will ssocite trigoometric series + cos() + b si(), or i terms of the epoetil e i, series of the form c e i. Z For most of the
More informationSTUDENT S COMPANIONS IN BASIC MATH: THE SECOND. Basic Identities in Algebra. Let us start with a basic identity in algebra:
STUDENT S COMPANIONS IN BASIC MATH: THE SECOND Bsic Idetities i Algebr Let us strt with bsic idetity i lgebr: 2 b 2 ( b( + b. (1 Ideed, multiplyig out the right hd side, we get 2 +b b b 2. Removig the
More informationMore About Expected Value and Variance
More Aout Expected Vlue d Vrice Pge of 5 E[ X ] Expected vlue,, hs umer of iterestig properties These re t likely to e used i this course eyod this lesso, ut my come ito ply i lter sttistics course Properties
More informationGeometric Sequences. Definition: A geometric sequence is a sequence of the form
Geometic equeces Aothe simple wy of geetig sequece is to stt with umbe d epetedly multiply it by fixed ozeo costt. This type of sequece is clled geometic sequece. Defiitio: A geometic sequece is sequece
More informationMath Bowl 2009 Written Test Solutions. 2 8i
Mth owl 009 Writte Test Solutios i? i i i i i ( i)( i ( i )( i ) ) 8i i i (i ) 9i 8 9i 9 i How my pirs of turl umers ( m, ) stisfy the equtio? m We hve to hve m d d, the m ; d, the 0 m m Tryig these umers,
More informationChapter 3 Section 3 Lesson Additional Rules for Exponents
Chpter Sectio Lesso Additiol Rules for Epoets Itroductio I this lesso we ll eie soe dditiol rules tht gover the behvior of epoets The rules should be eorized; they will be used ofte i the reiig chpters
More informationSecondary Math 2 Honors. Unit 2 Polynomials, Exponents, Radicals & Complex Numbers
Secodr Mth Hoors Uit Polomils, Epoets, Rdicls & Comple Numbers. Addig, Subtrctig, d Multiplig Polomils Notes Moomil: A epressio tht is umber, vrible, or umbers d vribles multiplied together. Moomils ol
More informationMATH 90 CHAPTER 5 Name:.
MATH 90 CHAPTER 5 Nme:. 5.1 Multiplictio of Expoets Need To Kow Recll expoets The ide of expoet properties Apply expoet properties Expoets Expoets me repeted multiplictio. 3 4 3 4 4 ( ) Expoet Properties
More informationIntro to Sequences / Arithmetic Sequences and Series Levels
Itro to Sequeces / Arithmetic Sequeces ad Series Levels Level : pg. 569: #7, 0, 33 Pg. 575: #, 7, 8 Pg. 584: #8, 9, 34, 36 Levels, 3, ad 4(Fiboacci Sequece Extesio) See Hadout Check for Uderstadig Level
More information8.2 Simplifying Radicals
. Simplifig Rdicls I the lst sectio we sw tht sice. However, otice tht (). So hs two differet squre roots. Becuse of this we eed to defie wht we cll the pricipl squre root so tht we c distiguish which
More informationm n Use technology to discover the rules for forms such as a a, various integer values of m and n and a fixed integer value a.
TIth.co Alger Expoet Rules ID: 988 Tie required 25 iutes Activity Overview This ctivity llows studets to work idepedetly to discover rules for workig with expoets, such s Multiplictio d Divisio of Like
More informationApplication: Volume. 6.1 Overture. Cylinders
Applictio: Volume 61 Overture I this chpter we preset other pplictio of the defiite itegrl, this time to fid volumes of certi solids As importt s this prticulr pplictio is, more importt is to recogize
More informationArithmetic Sequences
. Arithmetic Sequeces Essetial Questio How ca you use a arithmetic sequece to describe a patter? A arithmetic sequece is a ordered list of umbers i which the differece betwee each pair of cosecutive terms,
More informationwhen n = 1, 2, 3, 4, 5, 6, This list represents the amount of dollars you have after n days. Note: The use of is read as and so on.
Geometric eries Before we defie what is meat by a series, we eed to itroduce a related topic, that of sequeces. Formally, a sequece is a fuctio that computes a ordered list. uppose that o day 1, you have
More informations = 1 2 at2 + v 0 t + s 0
Mth A UCB, Sprig A. Ogus Solutios for Problem Set 4.9 # 5 The grph of the velocity fuctio of prticle is show i the figure. Sketch the grph of the positio fuctio. Assume s) =. A sketch is give below. Note
More informationModule 4: Mathematical Induction
Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate
More informationUNIT FIVE DETERMINANTS
UNIT FIVE DETERMINANTS. INTRODUTION I uit oe the determit of mtrix ws itroduced d used i the evlutio of cross product. I this chpter we exted the defiitio of determit to y size squre mtrix. The determit
More informationSequences and Series
Secto 9. Sequeces d Seres You c thk of sequece s fucto whose dom s the set of postve tegers. f ( ), f (), f (),... f ( ),... Defto of Sequece A fte sequece s fucto whose dom s the set of postve tegers.
More informationRiemann Sums y = f (x)
Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, oegative fuctio o the closed iterval [a, b] Fid
More informationARITHMETIC AND GEOMETRIC PROGRESSIONS
Arithmetic Ad Geometric Progressios Sequeces Ad ARITHMETIC AND GEOMETRIC PROGRESSIONS Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives
More informationReview for College Algebra Final Exam
Review for College Algebra Fial Exam (Please remember that half of the fial exam will cover chapters 14. This review sheet covers oly the ew material, from chapters 5 ad 7.) 5.1 Systems of equatios i
More informationSection 3.3: Geometric Sequences and Series
ectio 3.3: Geometic equeces d eies Geometic equeces Let s stt out with defiitio: geometic sequece: sequece i which the ext tem is foud by multiplyig the pevious tem by costt (the commo tio ) Hee e some
More informationYour grandmother and her financial counselor
Sectio 10. Arithmetic Sequeces 963 Objectives Sectio 10. Fid the commo differece for a arithmetic sequece. Write s of a arithmetic sequece. Use the formula for the geeral of a arithmetic sequece. Use the
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More informationMATHEMATICS SYLLABUS SECONDARY 7th YEAR
Europe Schools Office of the SecretryGeerl Pedgogicl developmet Uit Ref.: 201101D41e2 Orig.: DE MATHEMATICS SYLLABUS SECONDARY 7th YEAR Stdrd level 5 period/week course Approved y the Joit Techig
More informationIntroduction to Algorithms Chapter 3 Growth of Functions. How fast will your program run?
Itroductio to Algorithms Chpter 3 Growth of Fuctios 3  How fst will your progrm ru? The ruig time of your progrm will deped upo: The lgorithm The iput Your implemettio of the lgorithm i progrmmig lguge
More informationA. Description: A simple queueing system is shown in Fig. 161. Customers arrive randomly at an average rate of
Queueig Theory INTRODUCTION Queueig theory dels with the study of queues (witig lies). Queues boud i rcticl situtios. The erliest use of queueig theory ws i the desig of telehoe system. Alictios of queueig
More informationA proof of Goldbach's hypothesis that all even numbers greater than four are the sum of two primes.
A roof of Goldbch's hyothesis tht ll eve umbers greter th four re the sum of two rimes By Ket G Sliker Abstrct I this er I itroduce model which llows oe to rove Goldbchs hyothesis The model is roduced
More informationWinter Camp 2012 Sequences Alexander Remorov. Sequences. Alexander Remorov
Witer Camp 202 Sequeces Alexader Remorov Sequeces Alexader Remorov alexaderrem@gmail.com Warmup Problem : Give a positive iteger, cosider a sequece of real umbers a 0, a,..., a defied as a 0 = 2 ad =
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationSection IV.5: Recurrence Relations from Algorithms
Sectio IV.5: Recurrece Relatios from Algorithms Give a recursive algorithm with iput size, we wish to fid a Θ (best big O) estimate for its ru time T() either by obtaiig a explicit formula for T() or by
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More informationInfinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More informationGeometric Sequences and Series. Geometric Sequences. Definition of Geometric Sequence. such that. a2 4
3330_0903qxd /5/05 :3 AM Page 663 Sectio 93 93 Geometric Sequeces ad Series 663 Geometric Sequeces ad Series What you should lear Recogize, write, ad fid the th terms of geometric sequeces Fid th partial
More information4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then
SECTION 2.6 THE RATIO TEST 79 2.6. THE RATIO TEST We ow kow how to hadle series which we ca itegrate (the Itegral Test), ad series which are similar to geometric or pseries (the Compariso Test), but of
More informationArithmetic Sequences and Partial Sums. Arithmetic Sequences. Definition of Arithmetic Sequence. Example 1. 7, 11, 15, 19,..., 4n 3,...
3330_090.qxd 1/5/05 11:9 AM Page 653 Sectio 9. Arithmetic Sequeces ad Partial Sums 653 9. Arithmetic Sequeces ad Partial Sums What you should lear Recogize,write, ad fid the th terms of arithmetic sequeces.
More information2.7 Sequences, Sequences of Sets
2.7. SEQUENCES, SEQUENCES OF SETS 67 2.7 Sequeces, Sequeces of Sets 2.7.1 Sequeces Defiitio 190 (sequece Let S be some set. 1. A sequece i S is a fuctio f : K S where K = { N : 0 for some 0 N}. 2. For
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More informationLesson 12. Sequences and Series
Retur to List of Lessos Lesso. Sequeces ad Series A ifiite sequece { a, a, a,... a,...} ca be thought of as a list of umbers writte i defiite order ad certai patter. It is usually deoted by { a } =, or
More informationn Using the formula we get a confidence interval of 80±1.64
9.52 The professor of sttistics oticed tht the rks i his course re orlly distributed. He hs lso oticed tht his orig clss verge is 73% with stdrd devitio of 12% o their fil exs. His fteroo clsses verge
More information4.1 Sigma Notation and Riemann Sums
0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas
More informationif A S, then X \ A S, and if (A n ) n is a sequence of sets in S, then n A n S,
Lecture 5: Borel Sets Topologically, the Borel sets i a topological space are the σalgebra geerated by the ope sets. Oe ca build up the Borel sets from the ope sets by iteratig the operatios of complemetatio
More informationReleased Assessment Questions, 2015 QUESTIONS
Relesed Assessmet Questios, 15 QUESTIONS Grde 9 Assessmet of Mthemtis Ademi Red the istrutios elow. Alog with this ooklet, mke sure you hve the Aswer Booklet d the Formul Sheet. You my use y spe i this
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationExample 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook  Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly
More informationChapter Gaussian Elimination
Chapter 04.06 Gaussia Elimiatio After readig this chapter, you should be able to:. solve a set of simultaeous liear equatios usig Naïve Gauss elimiatio,. lear the pitfalls of the Naïve Gauss elimiatio
More informationSect Simplifying Radical Expressions. We can use our properties of exponents to establish two properties of radicals:
70 Sect 11.  Simplifyig Rdicl Epressios Cocept #1 Multiplictio d Divisio Properties of Rdicls We c use our properties of epoets to estlish two properties of rdicls: () 1/ 1/ 1/ & ( ) 1/ 1/ 1/ Multiplictio
More informationThe Euler Totient, the Möbius and the Divisor Functions
The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship
More informationSum of Exterior Angles of Polygons TEACHER NOTES
Sum of Exterior Agles of Polygos TEACHER NOTES Math Objectives Studets will determie that the iterior agle of a polygo ad a exterior agle of a polygo form a liear pair (i.e., the two agles are supplemetary).
More informationA Resource for Freestanding Mathematics Qualifications
A pie chrt shows how somethig is divided ito prts  it is good wy of showig the proportio (or frctio) of the dt tht is i ech ctegory. To drw pie chrt:. Fid the totl umer of items.. Fid how my degrees represet
More informationMath C067 Sampling Distributions
Math C067 Samplig Distributios Sample Mea ad Sample Proportio Richard Beigel Some time betwee April 16, 2007 ad April 16, 2007 Examples of Samplig A pollster may try to estimate the proportio of voters
More informationThe second difference is the sequence of differences of the first difference sequence, 2
Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for
More informationx(x 1)(x 2)... (x k + 1) = [x] k n+m 1
1 Coutig mappigs For every real x ad positive iteger k, let [x] k deote the fallig factorial ad x(x 1)(x 2)... (x k + 1) ( ) x = [x] k k k!, ( ) k = 1. 0 I the sequel, X = {x 1,..., x m }, Y = {y 1,...,
More informationFourier Series and the Wave Equation Part 2
Fourier Series ad the Wave Equatio Part There are two big ideas i our work this week. The first is the use of liearity to break complicated problems ito simple pieces. The secod is the use of the symmetries
More informationIntroduction to Hypothesis Testing
Itroductio to Hypothesis Testig I Cosumer Reports, April, 978, the results of tste test were reported. Cosumer Reports commeted, "we do't cosider this result to be sttisticlly sigifict." At the time, Miller
More informationExample 27.1 Draw a Venn diagram to show the relationship between counting numbers, whole numbers, integers, and rational numbers.
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
More informationBasic Elements of Arithmetic Sequences and Series
MA40S PRECALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic
More informationFourier Series (Lecture 13)
Fourier Series (Lecture 3) ody s Objectives: Studets will be ble to: ) Determie the Fourier Coefficiets for periodic sigl b) Fid the stedystte respose for system forced with geerl periodic forcig Rrely
More informationBasic Arithmetic TERMINOLOGY
Bsic Arithmetic TERMINOLOGY Absolute vlue: The distce of umber from zero o the umber lie. Hece it is the mgitude or vlue of umber without the sig Directed umbers: The set of itegers or whole umbers f,,,
More informationMathematics. Vectors. hsn.uk.net. Higher. Contents. Vectors 128 HSN23100
hsn.uk.net Higher Mthemtics UNIT 3 OUTCOME 1 Vectors Contents Vectors 18 1 Vectors nd Sclrs 18 Components 18 3 Mgnitude 130 4 Equl Vectors 131 5 Addition nd Subtrction of Vectors 13 6 Multipliction by
More informationParents Guide to helping your child with Higher Maths
Prets Guide to helpig your child with Higher Mths The essece of mthemtics is ot to mke simple thigs complicted, but to mke complicted thigs simple. S. Gudder Arithmetic is beig ble to cout up to twety
More informationAP CALCULUS FORMULA LIST. f x + x f x f x + h f x. b a
AP CALCULUS FORMULA LIST 1 Defiitio of e: e lim 1+ x if x 0 Asolute Vlue: x x if x < 0 Defiitio of the Derivtive: ( ) f x + x f x f x + h f x f '( x) lim f '( x) lim x x h h f ( + h) f ( ) f '( ) lim derivtive
More informationTrigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is
0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values
More informationChapter One BASIC MATHEMATICAL TOOLS
Chapter Oe BAIC MATHEMATICAL TOOL As the reader will see, the study of the time value of moey ivolves substatial use of variables ad umbers that are raised to a power. The power to which a variable is
More informationMaximum Likelihood Estimators.
Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio
More informationPrinciple of Mathematical Induction
Secto. Prcple of Mthemtcl Iducto.. Defto Mthemtcl ducto s techque of proof used to check ssertos or clms bout processes tht occur repettvely ccordg to set ptter. It s oe of the stdrd techques of proof
More informationDEPARTMENT OF ACTUARIAL STUDIES RESEARCH PAPER SERIES
DEPARTMENT OF ACTUARIAL STUDIES RESEARCH PAPER SERIES The ultibioil odel d pplictios by Ti Kyg Reserch Pper No. 005/03 July 005 Divisio of Ecooic d Ficil Studies Mcqurie Uiversity Sydey NSW 09 Austrli
More informationThe frequency relationship can also be written in term of logarithms
Pitch ad Frequecy by Mark Frech Departmet of Mechaical Egieerig Techology Purdue Uiversity rmfrech@purdue.edu There are two differet kids of scales importat to a guitar player, eve tempered ad diatoic.
More informationSection 9.2 Series and Convergence
Sectio 9. Series ad Covergece Goals of Chapter 9 Approximate Pi Prove ifiite series are aother importat applicatio of limits, derivatives, approximatio, slope, ad cocavity of fuctios. Fid challegig atiderivatives
More informationSEQUENCES AND SERIES CHAPTER
CHAPTER SEQUENCES AND SERIES Whe the Grat family purchased a computer for $,200 o a istallmet pla, they agreed to pay $00 each moth util the cost of the computer plus iterest had bee paid The iterest each
More informationGeometry Notes SIMILAR TRIANGLES
Similr Tringles Pge 1 of 6 SIMILAR TRIANGLES Objectives: After completing this section, you shoul be ble to o the following: Clculte the lengths of sies of similr tringles. Solve wor problems involving
More informationIntegration by Substitution
Integrtion by Substitution Dr. Philippe B. Lvl Kennesw Stte University August, 8 Abstrct This hndout contins mteril on very importnt integrtion method clled integrtion by substitution. Substitution is
More informationTHE GEOMETRIC SERIES
Mthemtics Revisio Guides The Geometic eies Pge of M.K. HOME TUITION Mthemtics Revisio Guides Level: A / A Level AQA : C Edexcel: C OCR: C OCR MEI: C THE GEOMETRIC ERIE Vesio :. Dte: 8060 Exmples 7 d
More informationSection 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult
More informationGray level image enhancement using the Bernstein polynomials
Buletiul Ştiiţiic l Uiersităţii "Politehic" di Timişor Seri ELECTRONICĂ şi TELECOMUNICAŢII TRANSACTIONS o ELECTRONICS d COMMUNICATIONS Tom 47(6), Fscicol , 00 Gry leel imge ehcemet usig the Berstei polyomils
More informationEven and Odd Functions
Eve d Odd Fuctios Beore lookig t urther emples o Fourier series it is useul to distiguish two clsses o uctios or which the Euler Fourier ormuls or the coeiciets c be simpliied. The two clsses re eve d
More informationMath : Sequences and Series
EPProgram  Strisuksa School  Roiet Math : Sequeces ad Series Dr.Wattaa Toutip  Departmet of Mathematics Kho Kae Uiversity 00 :Wattaa Toutip wattou@kku.ac.th http://home.kku.ac.th/wattou. Sequeces
More informationSample. Activity Library: Volume II. Activity Collections. Featuring realworld context collections:
Activity Library: Volume II Sample Activity Collectios Featurig realworld cotext collectios: Arithmetic II Fractios, Percets, Decimals III Fractios, Percets, Decimals IV Geometry I Geometry II Graphig
More informationChapter 9: Correlation and Regression: Solutions
Chapter 9: Correlatio ad Regressio: Solutios 9.1 Correlatio I this sectio, we aim to aswer the questio: Is there a relatioship betwee A ad B? Is there a relatioship betwee the umber of emploee traiig hours
More informationPresent and future value formulae for uneven cash flow Based on performance of a Business
Advces i Mgemet & Applied Ecoomics, vol., o., 20, 9309 ISSN: 7927544 (prit versio), 7927552 (olie) Itertiol Scietific Press, 20 Preset d future vlue formule for ueve csh flow Bsed o performce of Busiess
More informationHermitian Operators. Eigenvectors of a Hermitian operator. Definition: an operator is said to be Hermitian if it satisfies: A =A
Heriti Opertors Defiitio: opertor is sid to be Heriti if it stisfies: A A Altertively clled self doit I QM we will see tht ll observble properties st be represeted by Heriti opertors Theore: ll eigevles
More informationWe will begin this chapter with a quick refresher of what an exponent is.
.1 Exoets We will egi this chter with quick refresher of wht exoet is. Recll: So, exoet is how we rereset reeted ultilictio. We wt to tke closer look t the exoet. We will egi with wht the roerties re for
More informationEssential Question How can you use properties of exponents to simplify products and quotients of radicals?
. Properties of Ratioal Expoets ad Radicals Essetial Questio How ca you use properties of expoets to simplify products ad quotiets of radicals? Reviewig Properties of Expoets Work with a parter. Let a
More informationChapter 7. In the questions below, describe each sequence recursively. Include initial conditions and assume that the sequences begin with a 1.
Use the followig to aswer questios 6: Chapter 7 I the questios below, describe each sequece recursively Iclude iitial coditios ad assume that the sequeces begi with a a = 5 As: a = 5a,a = 5 The Fiboacci
More informationPolynomial Functions. Polynomial functions in one variable can be written in expanded form as ( )
Polynomil Functions Polynomil functions in one vrible cn be written in expnded form s n n 1 n 2 2 f x = x + x + x + + x + x+ n n 1 n 2 2 1 0 Exmples of polynomils in expnded form re nd 3 8 7 4 = 5 4 +
More informationG r a d e. 5 M a t h e M a t i c s. Patterns and relations
G r a d e 5 M a t h e M a t i c s Patters ad relatios Grade 5: Patters ad Relatios (Patters) (5.PR.1) Edurig Uderstadigs: Number patters ad relatioships ca be represeted usig variables. Geeral Outcome:
More information