In the case of a third degree polynomial we took the third difference and found them to be constants thus the polynomial difference holds.
|
|
- Eleanore Maxwell
- 5 years ago
- Views:
Transcription
1 Jso Mille 8 Udestd the piciples, popeties, d techiques elted to sequece, seies, summtio, d coutig sttegies d thei pplictios to polem solvig. Polomil Diffeece Theoem: f is polomil fuctio of degee iff fo set of -vlues tht foms ithmetic sequece, the th diffeeces of coespodig -vlues e equl d o-zeo. Emple: Coside the fuctio Y fo Sutctig the cosecutive vlues we see tht the diffeeces e equl 6 8 Sice is lie fuctio fist degee we ol hd to tke the st diffeeces. Howeve, s the degee of the polomil iceses, the diffeeces we must tke to get to costt must equl to the degee of the polomil. Emple: Coside the fuctio st diffeece d diffeece 6 8 d diffeece I the cse of thid degee polomil we took the thid diffeece d foud them to e costts thus the polomil diffeece holds. Fiite Diffeeces: Tke tle of -vlues coespodig to ithmetic sequece of -vlues, d compute the diffeeces of cosecutive -vlues. If those vlues of those diffeeces evetull ecome costt the the fuctio is polomil of degee equl to the ume of diffeeces tke to ech the costt.
2 Sequeces d Seies Aithmetic Lie Sequece: A sequece with costt diffeece, meig tht ech successive tem is lge tht the oe pecedig it. Tht is: >, d, o d, whee d is costt. Emple: Fo the sequece, 6, 9,, 5, 8, d d 6. The eplicit fomul is: -*. Aithmetic Seies: A sum of tems with commo diffeece d. The sum of the fist tems, epeseted t, is t. t o t [ d] Geometic Epoetil Sequece: A sequece with costt tio etwee tems fomed sttig with ume d multiplig ech successive tem costt. Tht is: > g * g, o whee is o-zeo costt. Emple: Fo the sequece,, 5, 5, 5, 5, d 5. The eplicit fomul is: 5 Geometic Seies: the sum of cosecutive tems of geometic sequece. This c e i foud with the followig fomul: iteges. i Fiocci Sequece: Nmed fte Leodo of Pis k Fiocci who studied the sequece d its popeties i the th cetu. It is foud i supisig ume of loctios i tue icludig piecoes, su flowes, disies, d eve sil shells Ech successive tem is foud ddig the two tems efoe it. The eplicit fomul to clculte the th tem 5 5 is 5 Howeve, the followig ecusive fomul is esie to use d is moe t the high school level., fo
3 Pemuttios d Comitios Pemuttio: igmet of ojects i specific ode. The ume of pemuttios of ojects is deoted! which is defied s:! Emple: ABC c e pemutted 6 ws : ABC, ACB, CBA, CAB, BAC, BCA! 6 Fudmetl Coutig Piciple: If thee e ws to do oe thig, d m ws to do othe, the thee e m* ws of doig oth. Emple: Suppose ou oll die d flip coi. Thee e outcomes fo flippig coi d 6 outcomes of ollig dice. B the Fudmetl Coutig Piciple thee e *6 ws of doig oth. Pemuttio of ojects tke t time: The ume of pemuttios of ojects! tke t time, deoted P,, o P whee.! Emple: Fid the ume of ws to ed diffeet ooks selected fom collectio of.!! 9! P! 9! 9! Comitio: A igmet of ojects i which ode does ot mtte.! Comitio of ojects tke t time:!! Emple: I collectio of ooks thee e 8 msteies d omce ovels. How m ws e thee to choose msteies d omce ovels, espectivel? 8 8! Selectig msteies 56!8!! Selectig omces 6!! Usig the fudmetl coutig piciple 56*6 6 possile ws.
4 Pscl s Tigle d Biomil Coefficiets The iomil epds to. If we look t iomils with othe epoetil powes we develop the followig ptte i the shpe of tigle. The fist d lst eties i ech ow e lws d ll othe eties e foud ddig the tems foud immeditel ove to the left d of it. It is esult of the epsio of. Emples: Pscl s tigle is es to use epesettio of the epsio of, d hs m popeties ll the powes of fom to occu i ode I ech tem, the epoets of d dd to If the powe of is, the the coefficiet of the tem is The th ow cotis eties. 5 The k th et i ow is C k- 6 The sum of ll the eties i ow equls Biomil Theoem: Emple: Epd 7 6 o
5 Polems:. Fom goup of smokes d 5 osmokes, uivesit eseche will choose 5 idividuls to pticipte i stud o lug cce.. How m ws c this e doe if it does ot mtte how m smoke o osmokes e icluded?. I how m ws c this e doe if ectl smokes must e chose? c. Fid the poilit tht ectl of the 5 people doml selected fom the goup of smokes.. Geometic figues e ofte epeseted the lettes tht idetif ech vete. Detemie the ume of ws tht ech figue elow c e med with the lettes A, B, C, D, E, d F. Assume tht ech figue is iegul.. Tigle. Qudiltel c. Which two shpes will hve the sme ume of possile mes?. I the figue elow, the sum of ech digol is t the ed of the digol.. Fid the missig sums.. Wht is the me of the sequece fomed the sums? Schultz, Ellis, Hollowell, Keed Wisto,, pp. 79???? How m sques e o 8 8 checkeod?. Let f the ume of sques o checkeod. Is f polomil fuctio? Justif ou swe.
6 5. Thee is sto ofte told out Kl Fiedich Guss. At ge he poved the Fudmetl Theoem of Alge. Whe he ws i thid gde, his clss misehved d the teche gve the followig polem s puishmet: Add the iteges fom to. It is sid tht Guss solved the polem i lmost o time t ll lettig S e the desied sum, usig the commuttive d ssocitive popeties to wite the sum i evese ode, d ddig the coespodig tems. The sums wee ll the sme, s, so S. Detemie wht is d wht Kl detemied the sum to e. Sek, 99, pp Suppose thete hs 6 sets i the fist ow d tht ech ow hs moe sets th the pevious ow. If thee e ows i the thete,. how m sets e i the lst ow?. how m sets e thee i ll? Sek, 99, pp Wite the seies usig Σ-ottio: c. the sum of the sques of the iteges fom to Sek, 99, pp A m stts with pi of its. How m pis of its c e poduced fom tht fist pi if the give ith to ew pi ech moth d it tkes moth fo pi to mtue d epoduce? LIVIO,
7 Aswes:.. 5 C 5 5,. C 5 C * 5 6 c.6/5,.7%! 6!.. P!!. 6 c. Petgo d Hego.. The missig umes e 5 8 d if the tigle ws cotiued This is the Fiocci sequece. Cout the ume of sques, up to 88 sques. dimesio of side of od: f # of sques: Yes f is d degee polomil ecuse the thid diffeeces e costt. f 6 5. Guss let S Usig the Commuttive d Associtive Popeties, the sum c e witte i evese ode: S9998. Addig the two togethe we get S times S* S55 So is d Kl detemied the sum to e To fid out how m sets e i the lst ow multipl the ume of ows the ume of sets eig dded ech ow sutctig tht e ot dded to the fist ow: *-. To fid the totl ume of sets i the thete usig the fomul: t [ d] t t t 5 5 [ 6 ] [ 68] i. 9 i 8 i i c. i i 8. Moth Pis
8 Refeeces Livio, M.. The Golde Rtio. New Yok: Rdom House. Pickove, C.A. 5. A Pssio fo Mthemtics. Hooke, NJ: Joh Wile & Sos Schultz, J.E., Ellis, W., Hollowell, K.A., & Keed, P.A.. Alge. New Yok: Holt, Rieht, Wisto. Sek, S. L., Et Al. 99. Advced Alge. Illiois: Scott Foesm. Wlpole, R.E., Mes, R.H., Mes, S. L., & Ye, K.. Poilit d Sttistics fo Egiees & Scietists. 7 th ed. Uppe Sddle Rive, NJ: Petice Hll.
PROGRESSION AND SERIES
INTRODUCTION PROGRESSION AND SERIES A gemet of umbes {,,,,, } ccodig to some well defied ule o set of ules is clled sequece Moe pecisely, we my defie sequece s fuctio whose domi is some subset of set of
More informationSummary: Binomial Expansion...! r. where
Summy: Biomil Epsio 009 M Teo www.techmejcmth-sg.wes.com ) Re-cp of Additiol Mthemtics Biomil Theoem... whee )!!(! () The fomul is ville i MF so studets do ot eed to memoise it. () The fomul pplies oly
More informationBINOMIAL THEOREM SOLUTION. 1. (D) n. = (C 0 + C 1 x +C 2 x C n x n ) (1+ x+ x 2 +.)
BINOMIAL THEOREM SOLUTION. (D) ( + + +... + ) (+ + +.) The coefficiet of + + + +... + fo. Moeove coefficiet of is + + + +... + if >. So. (B)... e!!!! The equied coefficiet coefficiet of i e -.!...!. (A),
More informationPLANCESS RANK ACCELERATOR
PLANCESS RANK ACCELERATOR MATHEMATICS FOR JEE MAIN & ADVANCED Sequeces d Seies 000questios with topic wise execises 000 polems of IIT-JEE & AIEEE exms of lst yes Levels of Execises ctegoized ito JEE Mi
More informationCHAPTER 5 : SERIES. 5.2 The Sum of a Series Sum of Power of n Positive Integers Sum of Series of Partial Fraction Difference Method
CHAPTER 5 : SERIES 5.1 Seies 5. The Sum of a Seies 5..1 Sum of Powe of Positive Iteges 5.. Sum of Seies of Patial Factio 5..3 Diffeece Method 5.3 Test of covegece 5.3.1 Divegece Test 5.3. Itegal Test 5.3.3
More informationConsider unordered sample of size r. This sample can be used to make r! Ordered samples (r! permutations). unordered sample
Uodeed Samples without Replacemet oside populatio of elemets a a... a. y uodeed aagemet of elemets is called a uodeed sample of size. Two uodeed samples ae diffeet oly if oe cotais a elemet ot cotaied
More informationEXERCISE - 01 CHECK YOUR GRASP
EXERISE - 0 HEK YOUR GRASP 3. ( + Fo sum of coefficiets put ( + 4 ( + Fo sum of coefficiets put ; ( + ( 4. Give epessio c e ewitte s 7 4 7 7 3 7 7 ( 4 3( 4... 7( 4 7 7 7 3 ( 4... 7( 4 Lst tem ecomes (4
More informationMATH Midterm Solutions
MATH 2113 - Midtem Solutios Febuay 18 1. A bag of mables cotais 4 which ae ed, 4 which ae blue ad 4 which ae gee. a How may mables must be chose fom the bag to guaatee that thee ae the same colou? We ca
More information,... are the terms of the sequence. If the domain consists of the first n positive integers only, the sequence is a finite sequence.
Chpter 9 & 0 FITZGERALD MAT 50/5 SECTION 9. Sequece Defiitio A ifiite sequece is fuctio whose domi is the set of positive itegers. The fuctio vlues,,, 4,...,,... re the terms of the sequece. If the domi
More information= 5! 3! 2! = 5! 3! (5 3)!. In general, the number of different groups of r items out of n items (when the order is ignored) is given by n!
0 Combiatoial Aalysis Copyight by Deiz Kalı 4 Combiatios Questio 4 What is the diffeece betwee the followig questio i How may 3-lette wods ca you wite usig the lettes A, B, C, D, E ii How may 3-elemet
More information2002 Quarter 1 Math 172 Final Exam. Review
00 Qute Mth 7 Fil Exm. Review Sectio.. Sets Repesettio of Sets:. Listig the elemets. Set-uilde Nottio Checkig fo Memeship (, ) Compiso of Sets: Equlity (=, ), Susets (, ) Uio ( ) d Itesectio ( ) of Sets
More informationEinstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,
MSS SEQUENCE AND SERIES CA SEQUENCE A sequece is fuctio of tul ubes with codoi is the set of el ubes (Coplex ubes. If Rge is subset of el ubes (Coplex ubes the it is clled el sequece (Coplex sequece. Exple
More informationAuchmuty High School Mathematics Department Sequences & Series Notes Teacher Version
equeces ad eies Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio A sequece takes the fom,,7,0,, while 7 0 is a seies. Thee ae two types of sequece/seies aithmetic ad geometic.
More informationCounting Functions and Subsets
CHAPTER 1 Coutig Fuctios ad Subsets This chapte of the otes is based o Chapte 12 of PJE See PJE p144 Hee ad below, the efeeces to the PJEccles book ae give as PJE The goal of this shot chapte is to itoduce
More informationME 501A Seminar in Engineering Analysis Page 1
Fobeius ethod pplied to Bessel s Equtio Octobe, 7 Fobeius ethod pplied to Bessel s Equtio L Cetto Mechicl Egieeig 5B Sei i Egieeig lsis Octobe, 7 Outlie Review idte Review lst lectue Powe seies solutios/fobeius
More informationBaltimore County ARML Team Formula Sheet, v2.1 (08 Apr 2008) By Raymond Cheong. Difference of squares Difference of cubes Sum of cubes.
Bltimoe Couty ARML Tem Fomul Seet, v. (08 Ap 008) By Rymo Ceog POLYNOMIALS Ftoig Diffeee of sques Diffeee of ues Sum of ues Ay itege O iteges ( )( ) 3 3 ( )( ) 3 3 ( )( ) ( )(... ) ( )(... ) Biomil expsio
More informationThe total number of permutations of S is n!. We denote the set of all permutations of S by
DETERMINNTS. DEFINITIONS Def: Let S {,,, } e the set of itegers from to, rrged i scedig order. rerrgemet jjj j of the elemets of S is clled permuttio of S. S. The totl umer of permuttios of S is!. We deote
More informationBINOMIAL THEOREM An expression consisting of two terms, connected by + or sign is called a
BINOMIAL THEOREM hapte 8 8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4 5y, etc., ae all biomial epessios. 8.. Biomial theoem If
More informationUNIT V: Z-TRANSFORMS AND DIFFERENCE EQUATIONS. Dr. V. Valliammal Department of Applied Mathematics Sri Venkateswara College of Engineering
UNIT V: -TRANSFORMS AND DIFFERENCE EQUATIONS D. V. Vllimml Deptmet of Applied Mthemtics Si Vektesw College of Egieeig TOPICS:. -Tsfoms Elemet popeties.. Ivese -Tsfom usig ptil fctios d esidues. Covolutio
More informationBINOMIAL THEOREM NCERT An expression consisting of two terms, connected by + or sign is called a
8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4, etc., ae all biomial 5y epessios. 8.. Biomial theoem BINOMIAL THEOREM If a ad b ae
More informationMathacle. PSet Stats, Concepts In Statistics Level Number Name: Date:
APPENDEX I. THE RAW ALGEBRA IN STATISTICS A I-1. THE INEQUALITY Exmple A I-1.1. Solve ech iequlity. Write the solutio i the itervl ottio..) 2 p - 6 p -8.) 2x- 3 < 5 Solutio:.). - 4 p -8 p³ 2 or pî[2, +
More informationWeek 8. Topic 2 Properties of Logarithms
Week 8 Topic 2 Popeties of Logithms 1 Week 8 Topic 2 Popeties of Logithms Intoduction Since the esult of ithm is n eponent, we hve mny popeties of ithms tht e elted to the popeties of eponents. They e
More informationUsing Counting Techniques to Determine Probabilities
Kowledge ticle: obability ad Statistics Usig outig Techiques to Detemie obabilities Tee Diagams ad the Fudametal outig iciple impotat aspect of pobability theoy is the ability to detemie the total umbe
More informationGraphing Review Part 3: Polynomials
Grphig Review Prt : Polomils Prbols Recll, tht the grph of f ( ) is prbol. It is eve fuctio, hece it is smmetric bout the bout the -is. This mes tht f ( ) f ( ). Its grph is show below. The poit ( 0,0)
More informationThe Area of a Triangle
The e of Tingle tkhlid June 1, 015 1 Intodution In this tile we will e disussing the vious methods used fo detemining the e of tingle. Let [X] denote the e of X. Using se nd Height To stt off, the simplest
More informationProgression. CATsyllabus.com. CATsyllabus.com. Sequence & Series. Arithmetic Progression (A.P.) n th term of an A.P.
Pogessio Sequece & Seies A set of umbes whose domai is a eal umbe is called a SEQUENCE ad sum of the sequece is called a SERIES. If a, a, a, a 4,., a, is a sequece, the the expessio a + a + a + a 4 + a
More informationx a y n + b = 1 0<b a, n > 0 (1.1) x 1 - a y = b 0<b a, n > 0 (1.1') b n sin 2 + cos 2 = 1 x n = = cos 2 6 Superellipse (Lamé curve)
6 Supeellipse (Lmé cuve) 6. Equtios of supeellipse A supeellipse (hoizotlly log) is epessed s follows. Implicit Equtio y + b 0 0 (.) Eplicit Equtio y b - 0 0 (.') Whe 3, b, the supeellipses fo
More informationUnit 1. Extending the Number System. 2 Jordan School District
Uit Etedig the Number System Jord School District Uit Cluster (N.RN. & N.RN.): Etedig Properties of Epoets Cluster : Etedig properties of epoets.. Defie rtiol epoets d eted the properties of iteger epoets
More informationWe show that every analytic function can be expanded into a power series, called the Taylor series of the function.
10 Lectue 8 We show tht evey lytic fuctio c be expded ito powe seies, clled the Tylo seies of the fuctio. Tylo s Theoem: Let f be lytic i domi D & D. The, f(z) c be expessed s the powe seies f( z) b (
More informationSOLUTIONS ( ) ( )! ( ) ( ) ( ) ( )! ( ) ( ) ( ) ( ) n r. r ( Pascal s equation ). n 1. Stepanov Dalpiaz
STAT UIU Pctice Poblems # SOLUTIONS Stepov Dlpiz The followig e umbe of pctice poblems tht my be helpful fo completig the homewo, d will liely be vey useful fo studyig fo ems...-.-.- Pove (show) tht. (
More informationFundamentals of Mathematics. Pascal s Triangle An Investigation March 20, 2008 Mario Soster
Fudmetls of Mthemtics Pscl s Trigle A Ivestigtio Mrch 0, 008 Mrio Soster Historicl Timelie A trigle showig the iomil coefficiets pper i Idi ook i the 0 th cetury I the th cetury Chiese mthemtici Yg Hui
More informationChapter 2 Infinite Series Page 1 of 9
Chpter Ifiite eries Pge of 9 Chpter : Ifiite eries ectio A Itroductio to Ifiite eries By the ed of this sectio you will be ble to uderstd wht is met by covergece d divergece of ifiite series recogise geometric
More information«A first lesson on Mathematical Induction»
Bcgou ifotio: «A fist lesso o Mtheticl Iuctio» Mtheticl iuctio is topic i H level Mthetics It is useful i Mtheticl copetitios t ll levels It hs bee coo sight tht stuets c out the poof b theticl iuctio,
More informationSULIT 3472/2. Rumus-rumus berikut boleh membantu anda menjawab soalan. Simbol-simbol yang diberi adalah yang biasa digunakan.
SULT 347/ Rumus-umus eikut oleh memtu d mejw sol. Simol-simol yg diei dlh yg is diguk. LGER. 4c x 5. log m log m log 9. T d. m m m 6. log = log m log 0. S d m m 3. 7. log m log m. S, m m logc 4. 8. log.
More informationM5. LTI Systems Described by Linear Constant Coefficient Difference Equations
5. LTI Systes Descied y Lie Costt Coefficiet Diffeece Equtios Redig teil: p.34-4, 245-253 3/22/2 I. Discete-Tie Sigls d Systes Up til ow we itoduced the Fouie d -tsfos d thei popeties with oly ief peview
More informationWeek 13 Notes: 1) Riemann Sum. Aim: Compute Area Under a Graph. Suppose we want to find out the area of a graph, like the one on the right:
Week 1 Notes: 1) Riem Sum Aim: Compute Are Uder Grph Suppose we wt to fid out the re of grph, like the oe o the right: We wt to kow the re of the red re. Here re some wys to pproximte the re: We cut the
More informationANSWER KEY PHYSICS. Workdone X
ANSWER KEY PHYSICS 6 6 6 7 7 7 9 9 9 0 0 0 CHEMISTRY 6 6 6 7 7 7 9 9 9 0 0 60 MATHEMATICS 6 66 7 76 6 6 67 7 77 7 6 6 7 7 6 69 7 79 9 6 70 7 0 90 PHYSICS F L l. l A Y l A ;( A R L L A. W = (/ lod etesio
More informationis an ordered list of numbers. Each number in a sequence is a term of a sequence. n-1 term
Mthemticl Ptters. Arithmetic Sequeces. Arithmetic Series. To idetify mthemticl ptters foud sequece. To use formul to fid the th term of sequece. To defie, idetify, d pply rithmetic sequeces. To defie rithmetic
More informationUsing Difference Equations to Generalize Results for Periodic Nested Radicals
Usig Diffeece Equatios to Geealize Results fo Peiodic Nested Radicals Chis Lyd Uivesity of Rhode Islad, Depatmet of Mathematics South Kigsto, Rhode Islad 2 2 2 2 2 2 2 π = + + +... Vieta (593) 2 2 2 =
More informationGRAPHING LINEAR EQUATIONS. Linear Equations. x l ( 3,1 ) _x-axis. Origin ( 0, 0 ) Slope = change in y change in x. Equation for l 1.
GRAPHING LINEAR EQUATIONS Qudrt II Qudrt I ORDERED PAIR: The first umer i the ordered pir is the -coordite d the secod umer i the ordered pir is the y-coordite. (, ) Origi ( 0, 0 ) _-is Lier Equtios Qudrt
More informationElementary Linear Algebra
Elemetry Lier Alger Ato & Rorres, th Editio Lecture Set Chpter : Systems of Lier Equtios & Mtrices Chpter Cotets Itroductio to System of Lier Equtios Gussi Elimitio Mtrices d Mtri Opertios Iverses; Rules
More informationBy the end of this section you will be able to prove the Chinese Remainder Theorem apply this theorem to solve simultaneous linear congruences
Chapte : Theoy of Modula Aithmetic 8 Sectio D Chiese Remaide Theoem By the ed of this sectio you will be able to pove the Chiese Remaide Theoem apply this theoem to solve simultaeous liea cogueces The
More informationAdvanced Higher Maths: Formulae
Advced Highe Mths: Fomule Advced Highe Mthemtics Gee (G): Fomule you solutely must memoise i ode to pss Advced Highe mths. Rememe you get o fomul sheet t ll i the em! Ame (A): You do t hve to memoise these
More informationIndices and Logarithms
the Further Mthemtics etwork www.fmetwork.org.uk V 7 SUMMARY SHEET AS Core Idices d Logrithms The mi ides re AQA Ed MEI OCR Surds C C C C Lws of idices C C C C Zero, egtive d frctiol idices C C C C Bsic
More informationName: Period: Date: 2.1 Rules of Exponents
SM NOTES Ne: Period: Dte:.1 Rules of Epoets The followig properties re true for ll rel ubers d b d ll itegers d, provided tht o deoitors re 0 d tht 0 0 is ot cosidered. 1 s epoet: 1 1 1 = e.g.) 7 = 7,
More informationFOURIER SERIES PART I: DEFINITIONS AND EXAMPLES. To a 2π-periodic 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 informationEinstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,
MB BINOMIAL THEOREM Biomial Epessio : A algebaic epessio which cotais two dissimila tems is called biomial epessio Fo eample :,,, etc / ( ) Statemet of Biomial theoem : If, R ad N, the : ( + ) = a b +
More information10.5 Power Series. In this section, we are going to start talking about power series. A power series is a series of the form
0.5 Power Series I the lst three sectios, we ve spet most of tht time tlkig bout how to determie if series is coverget or ot. Now it is time to strt lookig t some specific kids of series d we will evetully
More information10 Statistical Distributions Solutions
Communictions Engineeing MSc - Peliminy Reding 1 Sttisticl Distiutions Solutions 1) Pove tht the vince of unifom distiution with minimum vlue nd mximum vlue ( is ) 1. The vince is the men of the sques
More informationPROGRESSIONS AND SERIES
PROGRESSIONS AND SERIES A sequece is lso clled progressio. We ow study three importt types of sequeces: () The Arithmetic Progressio, () The Geometric Progressio, () The Hrmoic Progressio. Arithmetic Progressio.
More informationFor this purpose, we need the following result:
9 Lectue Sigulities of omplex Fuctio A poit is clled sigulity of fuctio f ( z ) if f ( z ) is ot lytic t the poit. A sigulity is clled isolted sigulity of f ( z ), if f ( z ) is lytic i some puctued disk
More informationAssessment Center Elementary Algebra Study Guide for the ACCUPLACER (CPT)
Assessmet Ceter Elemetr Alger Stud Guide for the ACCUPLACER (CPT) The followig smple questios re similr to the formt d cotet of questios o the Accuplcer Elemetr Alger test. Reviewig these smples will give
More informationThe rabbit reproduction model. Partitions. Leonardo Fibonacci. Solve in integers. Recurrences and Continued Fractions. x 1 + x x 5 = 40 x k rk;
V. Amchi D. Sleto Get Theoeticl Ies I Comute Sciece CS 5-5 Sig Lectue 8 Fe, Cegie Mello Uivesit Recueces Cotiue Fctios Solve i iteges + + + 5 = ; = + + + 5 = 5 ; 9 Ptitios Solve i iteges + + z = ; ; z
More informationPhysicsAndMathsTutor.com
PhysicsAdMthsTuto.com 5. () Show tht d y d PhysicsAdMthsTuto.com Jue 009 4 y = sec = 6sec 4sec. (b) Fid Tylo seies epsio of sec π i scedig powes of 4, up to d 3 π icludig the tem i 4. (6) (4) blk *M3544A08*
More informationSection IV.6: The Master Method and Applications
Sectio IV.6: The Mster Method d Applictios Defiitio IV.6.1: A fuctio f is symptoticlly positive if d oly if there exists rel umer such tht f(x) > for ll x >. A cosequece of this defiitio is tht fuctio
More information7.5-Determinants in Two Variables
7.-eteminnts in Two Vibles efinition of eteminnt The deteminnt of sque mti is el numbe ssocited with the mti. Eve sque mti hs deteminnt. The deteminnt of mti is the single ent of the mti. The deteminnt
More informationLimit of a function:
- Limit of fuctio: We sy tht f ( ) eists d is equl with (rel) umer L if f( ) gets s close s we wt to L if is close eough to (This defiitio c e geerlized for L y syig tht f( ) ecomes s lrge (or s lrge egtive
More information8.3 Sequences & Series: Convergence & Divergence
8.3 Sequeces & Series: Covergece & Divergece A sequece is simply list of thigs geerted by rule More formlly, sequece is fuctio whose domi is the set of positive itegers, or turl umbers,,,3,. The rge of
More informationSOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES
#A17 INTEGERS 9 2009), 181-190 SOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES Deick M. Keiste Depatmet of Mathematics, Pe State Uivesity, Uivesity Pak, PA 16802 dmk5075@psu.edu James A. Selles Depatmet
More informationTaylor Polynomials. The Tangent Line. (a, f (a)) and has the same slope as the curve y = f (x) at that point. It is the best
Tylor Polyomils Let f () = e d let p() = 1 + + 1 + 1 6 3 Without usig clcultor, evlute f (1) d p(1) Ok, I m still witig With little effort it is possible to evlute p(1) = 1 + 1 + 1 (144) + 6 1 (178) =
More informationEDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 2- ALGEBRAIC TECHNIQUES TUTORIAL 1 - PROGRESSIONS
EDEXCEL NATIONAL CERTIFICATE UNIT 8 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME - ALGEBRAIC TECHNIQUES TUTORIAL - PROGRESSIONS CONTENTS Be able to apply algebaic techiques Aithmetic pogessio (AP): fist
More informationDiscrete Mathematics I Tutorial 12
Discrete Mthemtics I Tutoril Refer to Chpter 4., 4., 4.4. For ech of these sequeces fid recurrece reltio stisfied by this sequece. (The swers re ot uique becuse there re ifiitely my differet recurrece
More informationSemiconductors materials
Semicoductos mteils Elemetl: Goup IV, Si, Ge Biy compouds: III-V (GAs,GSb, ISb, IP,...) IV-VI (PbS, PbSe, PbTe,...) II-VI (CdSe, CdTe,...) Tey d Qutey compouds: G x Al -x As, G x Al -x As y P -y III IV
More informationINFINITE SERIES. ,... having infinite number of terms is called infinite sequence and its indicated sum, i.e., a 1
Appedix A.. Itroductio As discussed i the Chpter 9 o Sequeces d Series, sequece,,...,,... hvig ifiite umber of terms is clled ifiite sequece d its idicted sum, i.e., + + +... + +... is clled ifite series
More informationa) The average (mean) of the two fractions is halfway between them: b) The answer is yes. Assume without loss of generality that p < r.
Solutios to MAML Olympiad Level 00. Factioated a) The aveage (mea) of the two factios is halfway betwee them: p ps+ q ps+ q + q s qs qs b) The aswe is yes. Assume without loss of geeality that p
More informationn 2 + 3n + 1 4n = n2 + 3n + 1 n n 2 = n + 1
Ifiite Series Some Tests for Divergece d Covergece Divergece Test: If lim u or if the limit does ot exist, the series diverget. + 3 + 4 + 3 EXAMPLE: Show tht the series diverges. = u = + 3 + 4 + 3 + 3
More informationThe Pigeonhole Principle 3.4 Binomial Coefficients
Discete M athematic Chapte 3: Coutig 3. The Pigeohole Piciple 3.4 Biomial Coefficiets D Patic Cha School of Compute Sciece ad Egieeig South Chia Uivesity of Techology Ageda Ch 3. The Pigeohole Piciple
More informationSTD: XI MATHEMATICS Total Marks: 90. I Choose the correct answer: ( 20 x 1 = 20 ) a) x = 1 b) x =2 c) x = 3 d) x = 0
STD: XI MATHEMATICS Totl Mks: 90 Time: ½ Hs I Choose the coect nswe: ( 0 = 0 ). The solution of is ) = b) = c) = d) = 0. Given tht the vlue of thid ode deteminnt is then the vlue of the deteminnt fomed
More informationChapter 5. The Riemann Integral. 5.1 The Riemann integral Partitions and lower and upper integrals. Note: 1.5 lectures
Chpter 5 The Riem Itegrl 5.1 The Riem itegrl Note: 1.5 lectures We ow get to the fudmetl cocept of itegrtio. There is ofte cofusio mog studets of clculus betwee itegrl d tiderivtive. The itegrl is (iformlly)
More information( ) D x ( s) if r s (3) ( ) (6) ( r) = d dr D x
SIO 22B, Rudnick dpted fom Dvis III. Single vile sttistics The next few lectues e intended s eview of fundmentl sttistics. The gol is to hve us ll speking the sme lnguge s we move to moe dvnced topics.
More informationClass Summary. be functions and f( D) , we define the composition of f with g, denoted g f by
Clss Summy.5 Eponentil Functions.6 Invese Functions nd Logithms A function f is ule tht ssigns to ech element D ectly one element, clled f( ), in. Fo emple : function not function Given functions f, g:
More informationMATH /19: problems for supervision in week 08 SOLUTIONS
MATH10101 2018/19: poblems fo supevisio i week 08 Q1. Let A be a set. SOLUTIONS (i Pove that the fuctio c: P(A P(A, defied by c(x A \ X, is bijective. (ii Let ow A be fiite, A. Use (i to show that fo each
More informationNorthwest High School s Algebra 2
Northwest High School s Algebr Summer Review Pcket 0 DUE August 8, 0 Studet Nme This pcket hs bee desiged to help ou review vrious mthemticl topics tht will be ecessr for our success i Algebr. Istructios:
More informationPhysicsAndMathsTutor.com
PhsicsAMthsTuto.com 6. The hpeol H hs equtio, whee e costts. The lie L hs equtio m c, whee m c e costts. Leve lk () Give tht L H meet, show tht the -cooites of the poits of itesectio e the oots of the
More informationEigenfunction Expansion. For a given function on the internal a x b the eigenfunction expansion of f(x):
Eigefuctio Epsio: For give fuctio o the iterl the eigefuctio epsio of f(): f ( ) cmm( ) m 1 Eigefuctio Epsio (Geerlized Fourier Series) To determie c s we multiply oth sides y Φ ()r() d itegrte: f ( )
More informationSimultaneous Estimation of Adjusted Rate of Two Factors Using Method of Direct Standardization
Biometics & Biosttistics Itetiol Joul Simulteous Estimtio of Adjusted Rte of Two Fctos Usig Method of Diect Stddiztio Astct This ppe pesets the use of stddiztio o djustmet of tes d tios i compig two popultios
More information* power rule: * fraction raised to negative exponent: * expanded power rule:
Mth 15 Iteredite Alger Stud Guide for E 3 (Chpters 7, 8, d 9) You use 3 5 ote crd (oth sides) d scietific clcultor. You re epected to kow (or hve writte o our ote crd) foruls ou eed. Thik out rules d procedures
More informationMATHS FOR ENGINEERS ALGEBRA TUTORIAL 8 MATHEMATICAL PROGRESSIONS AND SERIES
MATHS FOR ENGINEERS ALGEBRA TUTORIAL 8 MATHEMATICAL PROGRESSIONS AND SERIES O completio of this ttoial yo shold be able to do the followig. Eplai aithmetical ad geometic pogessios. Eplai factoial otatio
More informationCh 3.4 Binomial Coefficients. Pascal's Identit y and Triangle. Chapter 3.2 & 3.4. South China University of Technology
Disc ete Mathem atic Chapte 3: Coutig 3. The Pigeohole Piciple 3.4 Biomial Coefficiets D Patic Cha School of Compute Sciece ad Egieeig South Chia Uivesity of Techology Pigeohole Piciple Suppose that a
More informationPre-Calculus - Chapter 3 Sections Notes
Pre-Clculus - Chpter 3 Sectios 3.1-3.4- Notes Properties o Epoets (Review) 1. ( )( ) = + 2. ( ) =, (c) = 3. 0 = 1 4. - = 1/( ) 5. 6. c Epoetil Fuctios (Sectio 3.1) Deiitio o Epoetil Fuctios The uctio deied
More informationThe number of r element subsets of a set with n r elements
Popositio: is The umbe of elemet subsets of a set with elemets Poof: Each such subset aises whe we pick a fist elemet followed by a secod elemet up to a th elemet The umbe of such choices is P But this
More informationLesson 4 Linear Algebra
Lesso Lier Algebr A fmily of vectors is lierly idepedet if oe of them c be writte s lier combitio of fiitely my other vectors i the collectio. Cosider m lierly idepedet equtios i ukows:, +, +... +, +,
More informationIntroduction to Matrix Algebra
Itrodutio to Mtri Alger George H Olso, Ph D Dotorl Progrm i Edutiol Ledership Applhi Stte Uiversit Septemer Wht is mtri? Dimesios d order of mtri A p q dimesioed mtri is p (rows) q (olums) rr of umers,
More information( ) ( ) ( ) ( ) Solved Examples. JEE Main/Boards = The total number of terms in the expansion are 8.
Mathematics. Solved Eamples JEE Mai/Boads Eample : Fid the coefficiet of y i c y y Sol: By usig fomula of fidig geeal tem we ca easily get coefficiet of y. I the biomial epasio, ( ) th tem is c T ( y )
More informationSM2H. Unit 2 Polynomials, Exponents, Radicals & Complex Numbers Notes. 3.1 Number Theory
SMH Uit Polyomils, Epoets, Rdicls & Comple Numbers Notes.1 Number Theory .1 Addig, Subtrctig, d Multiplyig Polyomils Notes Moomil: A epressio tht is umber, vrible, or umbers d vribles multiplied together.
More informationChapter 7. Kleene s Theorem. 7.1 Kleene s Theorem. The following theorem is the most important and fundamental result in the theory of FA s:
Chpte 7 Kleene s Theoem 7.1 Kleene s Theoem The following theoem is the most impotnt nd fundmentl esult in the theoy of FA s: Theoem 6 Any lnguge tht cn e defined y eithe egul expession, o finite utomt,
More informationApproximate Integration
Study Sheet (7.7) Approimte Itegrtio I this sectio, we will ler: How to fid pproimte vlues of defiite itegrls. There re two situtios i which it is impossile to fid the ect vlue of defiite itegrl. Situtio:
More informationPOWER SERIES R. E. SHOWALTER
POWER SERIES R. E. SHOWALTER. sequeces We deote by lim = tht the limit of the sequece { } is the umber. By this we me tht for y ε > 0 there is iteger N such tht < ε for ll itegers N. This mkes precise
More informationRiemann Integral and Bounded function. Ng Tze Beng
Riem Itegrl d Bouded fuctio. Ng Tze Beg I geerlistio of re uder grph of fuctio, it is ormlly ssumed tht the fuctio uder cosidertio e ouded. For ouded fuctio, the rge of the fuctio is ouded d hece y suset
More informationICS141: Discrete Mathematics for Computer Science I
Uivesity of Hawaii ICS141: Discete Mathematics fo Compute Sciece I Dept. Ifomatio & Compute Sci., Uivesity of Hawaii Ja Stelovsy based o slides by D. Bae ad D. Still Oigials by D. M. P. Fa ad D. J.L. Goss
More informationChapter 7 Infinite Series
MA Ifiite Series Asst.Prof.Dr.Supree Liswdi Chpter 7 Ifiite Series Sectio 7. Sequece A sequece c be thought of s list of umbers writte i defiite order:,,...,,... 2 The umber is clled the first term, 2
More informationZero Level Binomial Theorem 04
Zeo Level Biomial Theoem 0 Usig biomial theoem, epad the epasios of the Fid the th tem fom the ed i the epasio of followig : (i ( (ii, 0 Fid the th tem fom the ed i the epasio of (iii ( (iv ( a (v ( (vi,
More informationPhysicsAndMathsTutor.com
PhysicsAdMthsTuto.com PhysicsAdMthsTuto.com Jue 009 7. () Sketch the gph of y, whee >, showig the coodites of the poits whee the gph meets the es. () Leve lk () Solve, >. (c) Fid the set of vlues of fo
More informationPhysicsAndMathsTutor.com
PhysicsAMthsTuto.com . M 6 0 7 0 Leve lk 6 () Show tht 7 is eigevlue of the mti M fi the othe two eigevlues of M. (5) () Fi eigevecto coespoig to the eigevlue 7. *M545A068* (4) Questio cotiue Leve lk *M545A078*
More informationAdvanced Higher Maths: Formulae
: Fomule Gee (G): Fomule you bsolutely must memoise i ode to pss Advced Highe mths. Remembe you get o fomul sheet t ll i the em! Ambe (A): You do t hve to memoise these fomule, s it is possible to deive
More informationFig. 1. I a. V ag I c. I n. V cg. Z n Z Y. I b. V bg
ymmetricl Compoets equece impedces Although the followig focuses o lods, the results pply eqully well to lies, or lies d lods. Red these otes together with sectios.6 d.9 of text. Cosider the -coected lced
More informationBINOMIAL THEOREM & ITS SIMPLE APPLICATION
Etei lasses, Uit No. 0, 0, Vadhma Rig Road Plaza, Vikas Pui Et., Oute Rig Road New Delhi 0 08, Ph. : 9690, 87 MB Sllabus : BINOMIAL THEOREM & ITS SIMPLE APPLIATION Biomia Theoem fo a positive itegal ide;
More informationI. Exponential Function
MATH & STAT Ch. Eoetil Fuctios JCCSS I. Eoetil Fuctio A. Defiitio f () =, whee ( > 0 ) d is the bse d the ideedet vible is the eoet. [ = 1 4 4 4L 4 ] ties (Resf () = is owe fuctio i which the bse is the
More informationEXAMPLES. Leader in CBSE Coaching. Solutions of BINOMIAL THEOREM A.V.T.E. by AVTE (avte.in) Class XI
avtei EXAMPLES Solutios of AVTE by AVTE (avtei) lass XI Leade i BSE oachig 1 avtei SHORT ANSWER TYPE 1 Fid the th tem i the epasio of 1 We have T 1 1 1 1 1 1 1 1 1 1 Epad the followig (1 + ) 4 Put 1 y
More informationInterpolation. 1. What is interpolation?
Iterpoltio. Wht is iterpoltio? A uctio is ote give ol t discrete poits such s:.... How does oe id the vlue o t other vlue o? Well cotiuous uctio m e used to represet the + dt vlues with pssig through the
More information