EDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 2- ALGEBRAIC TECHNIQUES TUTORIAL 1 - PROGRESSIONS

Size: px
Start display at page:

Download "EDEXCEL NATIONAL CERTIFICATE UNIT 28 FURTHER MATHEMATICS FOR TECHNICIANS OUTCOME 2- ALGEBRAIC TECHNIQUES TUTORIAL 1 - PROGRESSIONS"

Transcription

1 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 tem (a), commo diffeece (d), th tem e.g. a +( )d ; aithmetic seies e.g. sum to tems, S a ) d Geometic pogessio (GP): fist tem (a), commo atio (), th tem e.g. a - ; a a geometic seies e.g. sum to tems, S, sum to ifiity S solutio of pactical poblems e.g. compoud iteest, age of speeds o a dillig machie Complex umbes: additio, subtactio, multiplicatio of a complex umbe i Catesia fom, vecto epesetatio of complex umbes, modulus ad agumet, pola epesetatio of complex umbes, multiplicatio ad divisio of complex umbes i pola fom, pola to Catesia fom ad vice vesa, use of calculato Statistical techiques: eview of measue of cetal tedecy, mea, stadad deviatio fo ugouped ad gouped data (equal itevals oly), vaiace It is assumed that the studet has completed the module MATHEMATICS FOR TECHNICIANS. D.J.Du

2 . ARITHMETIC PROGRESSION Coside the sequece of umbes 3,5,7,9,,. This is a pogessio that stats at 3 ad is iceased by each time so we say the commo diffeece is d = ad the statig value is a = 3. The sum of tems is called a ARITHMETIC SERIES. To fid this we would have to add them all up. The sum S would be: S a ( a d) a d d a d d d... a d d... a d a 3d ( a 4d)... a d d S a ( a d) It ca be show that the aswe is () x (aveage value). The aveage value is foud by addig the fist ad last tem ad dividig by so we ca wite: S a a d a S d a d S S a ) d This is the geeally accepted fomula fo evaluatig a pogessio. WORKED EXAMPLE No. Show that the fomula deived above gives the coect aswe fo the sum of the pogessio: Simply addig them up gives S = 5 Fo this pogessio = 5, d = ad a = S a ) d Usig the fomula gives WORKED EXAMPLE No. Evaluate the fist 0 tems of the pogessio a = 4 (the fist value) d = 4 (the diffeece betwee each umbe) = 0 Usig the fomula gives S a ) d 0 S S D.J.Du

3 WORKED EXAMPLE No. 3 A maufactue of steel bas oly sells them i batches of 0 m ad calculates the chage to customes usig the followig system. 0-0 m 0-0 m 0-30 m 00 ad so o with a maximum ode of 00 m Wite out the aithmetic pogessio ad the fomula fo calculatig the cost of a ode. Calculate the chage fo 00 m The pogessio is 0, 0, 00. The cost of batches is : S a ) d Fo 00 m = 0 a = 0 d = -0 0 S a ) d ()(0) 0 0 S The chage is 500 SELF ASSESSMENT EXERCISE No.. Evaluate the fist 5 tems of the pogessio Evaluate the fist 30 tems of the pogessio Evaluate the fist 0 tems of the pogessio How may tems ae equied i questio () to poduce a aswe of zeo? 5. Evaluate the fist 5 tems of the pogessio give i (). 6. A maufactue of a mass poduced poduct chages by the followig method. Quatity Pice ad so o with a maximum ode of 000 How much would a ode of 000 cost? (Aswe 390) (Aswe 570) (Aswe 00) (Aswe ) (Aswe 350) (Aswe 0) D.J.Du 3

4 . GEOMETRIC PROGRESSION A geometic pogessio is a seies of umbes statig at a ad the ext tem is the pevious tem multiplied by a commo atio. It follows that the pogessio is a, a, a, a 3..a If we add tems togethe we have a GEOMETRIC SERIES. Fo example let a =. The two gaphs show the affect of havig slightly lage ad slightly smalle tha. Whe is smalle tha the gaph shows that S eaches a fial value but whe is lage tha, S keeps gowig. The sum of tems would be give by: S = a + a + a + a 3.+ a S = a( ) multiply both sides by ( - ) ( - )S = a( ) ( - ) ( - )S = a{( ) - ( )} ( - )S = a{ - + )} a S Note this does ot wok if = Sometimes we wat to kow the value whe is ifiity. This is ot always possible but ofte the tems will geeally get smalle ad smalle ad covege o a value. This oly happes if the commo atio is less tha ad lage tha -. a Whe this happes the sum is S The poof is ot give hee. WORKED EXAMPLE No. 4 Fid the sum of the fist 6 tems of the geometic pogessio 5, 5(), 5(), 5() 3..5() a = 5, = ad = 6 6 a S a D.J.Du 4

5 WORKED EXAMPLE No. 5 Fid the sum of the fist 4 tems of the geometic pogessio 6, 6(3), 6(3), 6(3) 3..6(3) a = 6, = 3 ad = 4 4 a S a WORKED EXAMPLE No. 6 Fid the sum to ifiity of the seies, ( ½), (½ ), (½) 3..(½) a =, = ½ 4 a S a 4 / / SELF ASSESSMENT EXERCISE No.. Evaluate the fist 0 tems of the seies 3 + 3x + 3x + 3x 3 whe x = (Aswe 64). Evaluate the maximum value of the seies 0 + 0x + 0x + 0x 3 whe x = ½ (Aswe 40) 3. Evaluate the maximum value of the seies 0 + 0x + 0x + 0x 3 whe x = -½ (Aswe 3.333) D.J.Du 5

MATHS FOR ENGINEERS ALGEBRA TUTORIAL 8 MATHEMATICAL PROGRESSIONS AND SERIES

MATHS 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 information

Progression. CATsyllabus.com. CATsyllabus.com. Sequence & Series. Arithmetic Progression (A.P.) n th term of an A.P.

Progression. 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 information

Auchmuty High School Mathematics Department Sequences & Series Notes Teacher Version

Auchmuty 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 information

CHAPTER 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. 5.2 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 information

By the end of this section you will be able to prove the Chinese Remainder Theorem apply this theorem to solve simultaneous linear congruences

By 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 information

PROGRESSION AND SERIES

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 information

LESSON 15: COMPOUND INTEREST

LESSON 15: COMPOUND INTEREST High School: Expoeial Fuctios LESSON 15: COMPOUND INTEREST 1. You have see this fomula fo compoud ieest. Paamete P is the picipal amou (the moey you stat with). Paamete is the ieest ate pe yea expessed

More information

2012 GCE A Level H2 Maths Solution Paper Let x,

2012 GCE A Level H2 Maths Solution Paper Let x, GCE A Level H Maths Solutio Pape. Let, y ad z be the cost of a ticet fo ude yeas, betwee ad 5 yeas, ad ove 5 yeas categoies espectively. 9 + y + 4z =. 7 + 5y + z = 8. + 4y + 5z = 58.5 Fo ude, ticet costs

More information

a) The average (mean) of the two fractions is halfway between them: b) The answer is yes. Assume without loss of generality that p < r.

a) 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 information

BINOMIAL THEOREM An expression consisting of two terms, connected by + or sign is called a

BINOMIAL 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 information

Advanced Higher Formula List

Advanced Higher Formula List Advaced Highe Fomula List Note: o fomulae give i eam emembe eveythig! Uit Biomial Theoem Factoial! ( ) ( ) Biomial Coefficiet C!! ( )! Symmety Idetity Khayyam-Pascal Idetity Biomial Theoem ( y) C y 0 0

More information

BINOMIAL THEOREM NCERT An expression consisting of two terms, connected by + or sign is called a

BINOMIAL 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 information

Chapter 2 Sampling distribution

Chapter 2 Sampling distribution [ 05 STAT] Chapte Samplig distibutio. The Paamete ad the Statistic Whe we have collected the data, we have a whole set of umbes o desciptios witte dow o a pape o stoed o a compute file. We ty to summaize

More information

Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,

Einstein 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 information

MATH Midterm Solutions

MATH 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

Using Counting Techniques to Determine Probabilities

Using 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 information

MATH /19: problems for supervision in week 08 SOLUTIONS

MATH /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 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!

= 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 information

CfE Advanced Higher Mathematics Learning Intentions and Success Criteria BLOCK 1 BLOCK 2 BLOCK 3

CfE Advanced Higher Mathematics Learning Intentions and Success Criteria BLOCK 1 BLOCK 2 BLOCK 3 Eempla Pape Specime Pape Pages Eempla Pape Specime Pape Pages Eempla Pape Specime Pape Pages Abedee Gamma School Mathematics Depatmet CfE Advaced Highe Mathematics Leaig Itetios ad Success Citeia BLOCK

More information

Finite q-identities related to well-known theorems of Euler and Gauss. Johann Cigler

Finite q-identities related to well-known theorems of Euler and Gauss. Johann Cigler Fiite -idetities elated to well-ow theoems of Eule ad Gauss Joha Cigle Faultät fü Mathemati Uivesität Wie A-9 Wie, Nodbegstaße 5 email: oha.cigle@uivie.ac.at Abstact We give geealizatios of a fiite vesio

More information

Math III Final Exam Review. Name. Unit 1 Statistics. Definitions Population: Sample: Statistics: Parameter: Methods for Collecting Data Survey:

Math III Final Exam Review. Name. Unit 1 Statistics. Definitions Population: Sample: Statistics: Parameter: Methods for Collecting Data Survey: Math III Fial Exam Review Name Uit Statistics Defiitios Populatio: Sample: Statistics: Paamete: Methods fo Collectig Data Suvey: Obsevatioal Study: Expeimet: Samplig Methods Radom: Statified: Systematic:

More information

Counting Functions and Subsets

Counting 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 information

The Pigeonhole Principle 3.4 Binomial Coefficients

The 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 information

6.3 Testing Series With Positive Terms

6.3 Testing Series With Positive Terms 6.3. TESTING SERIES WITH POSITIVE TERMS 307 6.3 Testig Series With Positive Terms 6.3. Review of what is kow up to ow I theory, testig a series a i for covergece amouts to fidig the i= sequece of partial

More information

AIEEE 2004 (MATHEMATICS)

AIEEE 2004 (MATHEMATICS) AIEEE 004 (MATHEMATICS) Impotat Istuctios: i) The test is of hous duatio. ii) The test cosists of 75 questios. iii) The maimum maks ae 5. iv) Fo each coect aswe you will get maks ad fo a wog aswe you will

More information

Conditional Convergence of Infinite Products

Conditional Convergence of Infinite Products Coditioal Covegece of Ifiite Poducts William F. Tech Ameica Mathematical Mothly 106 1999), 646-651 I this aticle we evisit the classical subject of ifiite poducts. Fo stadad defiitios ad theoems o this

More information

Topic 1 2: Sequences and Series. A sequence is an ordered list of numbers, e.g. 1, 2, 4, 8, 16, or

Topic 1 2: Sequences and Series. A sequence is an ordered list of numbers, e.g. 1, 2, 4, 8, 16, or Topic : Sequeces ad Series A sequece is a ordered list of umbers, e.g.,,, 8, 6, or,,,.... A series is a sum of the terms of a sequece, e.g. + + + 8 + 6 + or... Sigma Notatio b The otatio f ( k) is shorthad

More information

Zero Level Binomial Theorem 04

Zero 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 information

Infinite Series. Definition. An infinite series is an expression of the form. Where the numbers u k are called the terms of the series.

Infinite Series. Definition. An infinite series is an expression of the form. Where the numbers u k are called the terms of the series. Ifiite Series Defiitio. A ifiite series is a expressio of the form uk = u + u + u + + u + () 2 3 k Where the umbers u k are called the terms of the series. Such a expressio is meat to be the result of

More information

Using Difference Equations to Generalize Results for Periodic Nested Radicals

Using 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 information

physicsandmathstutor.com

physicsandmathstutor.com physicsadmathstuto.com 2. Solve (a) 5 = 8, givig you aswe to 3 sigificat figues, (b) log 2 ( 1) log 2 = log 2 7. (3) (3) 4 *N23492B0428* 3. (i) Wite dow the value of log 6 36. (ii) Epess 2 log a 3 log

More information

KEY. Math 334 Midterm II Fall 2007 section 004 Instructor: Scott Glasgow

KEY. Math 334 Midterm II Fall 2007 section 004 Instructor: Scott Glasgow KEY Math 334 Midtem II Fall 7 sectio 4 Istucto: Scott Glasgow Please do NOT wite o this exam. No cedit will be give fo such wok. Rathe wite i a blue book, o o you ow pape, pefeably egieeig pape. Wite you

More information

SEQUENCES AND SERIES

SEQUENCES AND SERIES 9 SEQUENCES AND SERIES INTRODUCTION Sequeces have may importat applicatios i several spheres of huma activities Whe a collectio of objects is arraged i a defiite order such that it has a idetified first

More information

Chapter 8 Complex Numbers

Chapter 8 Complex Numbers Chapte 8 Complex Numbes Motivatio: The ae used i a umbe of diffeet scietific aeas icludig: sigal aalsis, quatum mechaics, elativit, fluid damics, civil egieeig, ad chaos theo (factals. 8.1 Cocepts - Defiitio

More information

THE ANALYTIC LARGE SIEVE

THE ANALYTIC LARGE SIEVE THE ANALYTIC LAGE SIEVE 1. The aalytic lage sieve I the last lectue we saw how to apply the aalytic lage sieve to deive a aithmetic fomulatio of the lage sieve, which we applied to the poblem of boudig

More information

Ch 3.4 Binomial Coefficients. Pascal's Identit y and Triangle. Chapter 3.2 & 3.4. South China University of Technology

Ch 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 information

VERY SHORT ANSWER TYPE QUESTIONS (1 MARK)

VERY SHORT ANSWER TYPE QUESTIONS (1 MARK) VERY SHORT ANSWER TYPE QUESTIONS ( MARK). If th term of a A.P. is 6 7 the write its 50 th term.. If S = +, the write a. Which term of the sequece,, 0, 7,... is 6? 4. If i a A.P. 7 th term is 9 ad 9 th

More information

Sect 5.3 Proportions

Sect 5.3 Proportions Sect 5. Proportios Objective a: Defiitio of a Proportio. A proportio is a statemet that two ratios or two rates are equal. A proportio takes a form that is similar to a aalogy i Eglish class. For example,

More information

CfE Advanced Higher Mathematics Course materials Topic 5: Binomial theorem

CfE Advanced Higher Mathematics Course materials Topic 5: Binomial theorem SCHOLAR Study Guide CfE Advaced Highe Mathematics Couse mateials Topic : Biomial theoem Authoed by: Fioa Withey Stilig High School Kae Withey Stilig High School Reviewed by: Magaet Feguso Peviously authoed

More information

Math 166 Week-in-Review - S. Nite 11/10/2012 Page 1 of 5 WIR #9 = 1+ r eff. , where r. is the effective interest rate, r is the annual

Math 166 Week-in-Review - S. Nite 11/10/2012 Page 1 of 5 WIR #9 = 1+ r eff. , where r. is the effective interest rate, r is the annual Math 66 Week-i-Review - S. Nite // Page of Week i Review #9 (F-F.4, 4.-4.4,.-.) Simple Iteest I = Pt, whee I is the iteest, P is the picipal, is the iteest ate, ad t is the time i yeas. P( + t), whee A

More information

On ARMA(1,q) models with bounded and periodically correlated solutions

On ARMA(1,q) models with bounded and periodically correlated solutions Reseach Repot HSC/03/3 O ARMA(,q) models with bouded ad peiodically coelated solutios Aleksade Weo,2 ad Agieszka Wy oma ska,2 Hugo Steihaus Cete, Woc aw Uivesity of Techology 2 Istitute of Mathematics,

More information

SOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES

SOME 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 information

Chapter 3: Theory of Modular Arithmetic 38

Chapter 3: Theory of Modular Arithmetic 38 Chapte 3: Theoy of Modula Aithmetic 38 Section D Chinese Remainde Theoem By the end of this section you will be able to pove the Chinese Remainde Theoem apply this theoem to solve simultaneous linea conguences

More information

At the end of this topic, students should be able to understand the meaning of finite and infinite sequences and series, and use the notation u

At the end of this topic, students should be able to understand the meaning of finite and infinite sequences and series, and use the notation u Natioal Jio College Mathematics Depatmet 00 Natioal Jio College 00 H Mathematics (Seio High ) Seqeces ad Seies (Lecte Notes) Topic : Seqeces ad Seies Objectives: At the ed of this topic, stdets shold be

More information

Disjoint Sets { 9} { 1} { 11} Disjoint Sets (cont) Operations. Disjoint Sets (cont) Disjoint Sets (cont) n elements

Disjoint Sets { 9} { 1} { 11} Disjoint Sets (cont) Operations. Disjoint Sets (cont) Disjoint Sets (cont) n elements Disjoit Sets elemets { x, x, } X =, K Opeatios x Patitioed ito k sets (disjoit sets S, S,, K Fid-Set(x - etu set cotaiig x Uio(x,y - make a ew set by combiig the sets cotaiig x ad y (destoyig them S k

More information

SEQUENCE AND SERIES NCERT

SEQUENCE AND SERIES NCERT 9. Overview By a sequece, we mea a arragemet of umbers i a defiite order accordig to some rule. We deote the terms of a sequece by a, a,..., etc., the subscript deotes the positio of the term. I view of

More information

Measures of Spread: Standard Deviation

Measures of Spread: Standard Deviation Measures of Spread: Stadard Deviatio So far i our study of umerical measures used to describe data sets, we have focused o the mea ad the media. These measures of ceter tell us the most typical value of

More information

1. Hilbert s Grand Hotel. The Hilbert s Grand Hotel has infinite many rooms numbered 1, 2, 3, 4

1. Hilbert s Grand Hotel. The Hilbert s Grand Hotel has infinite many rooms numbered 1, 2, 3, 4 . Hilbert s Grad Hotel The Hilbert s Grad Hotel has ifiite may rooms umbered,,,.. Situatio. The Hotel is full ad a ew guest arrives. Ca the mager accommodate the ew guest? - Yes, he ca. There is a simple

More information

Activity 3: Length Measurements with the Four-Sided Meter Stick

Activity 3: Length Measurements with the Four-Sided Meter Stick Activity 3: Legth Measuremets with the Four-Sided Meter Stick OBJECTIVE: The purpose of this experimet is to study errors ad the propagatio of errors whe experimetal data derived usig a four-sided meter

More information

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Statistics

ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER / Statistics ACCESS TO SCIENCE, ENGINEERING AND AGRICULTURE: MATHEMATICS 1 MATH00030 SEMESTER 1 018/019 DR. ANTHONY BROWN 8. Statistics 8.1. Measures of Cetre: Mea, Media ad Mode. If we have a series of umbers the

More information

10.6 ALTERNATING SERIES

10.6 ALTERNATING SERIES 0.6 Alteratig Series Cotemporary Calculus 0.6 ALTERNATING SERIES I the last two sectios we cosidered tests for the covergece of series whose terms were all positive. I this sectio we examie series whose

More information

Student s Name : Class Roll No. ADDRESS: R-1, Opp. Raiway Track, New Corner Glass Building, Zone-2, M.P. NAGAR, Bhopal

Student s Name : Class Roll No. ADDRESS: R-1, Opp. Raiway Track, New Corner Glass Building, Zone-2, M.P. NAGAR, Bhopal FREE Dowload Stud Package fom website: wwwtekoclassescom fo/u fopkj Hkh# tu] ugha vkjehks dke] foif s[k NksMs qja e/;e eu dj ';kea iq#"k flag ladyi dj] lgs foif vusd] ^cuk^ u NksMs /;s; dks] j?kqcj jk[ks

More information

APPENDIX F Complex Numbers

APPENDIX F Complex Numbers APPENDIX F Complex Numbers Operatios with Complex Numbers Complex Solutios of Quadratic Equatios Polar Form of a Complex Number Powers ad Roots of Complex Numbers Operatios with Complex Numbers Some equatios

More information

is also known as the general term of the sequence

is also known as the general term of the sequence Lesso : Sequeces ad Series Outlie Objectives: I ca determie whether a sequece has a patter. I ca determie whether a sequece ca be geeralized to fid a formula for the geeral term i the sequece. I ca determie

More information

Technical Report: Bessel Filter Analysis

Technical Report: Bessel Filter Analysis Sasa Mahmoodi 1 Techical Repot: Bessel Filte Aalysis 1 School of Electoics ad Compute Sciece, Buildig 1, Southampto Uivesity, Southampto, S17 1BJ, UK, Email: sm3@ecs.soto.ac.uk I this techical epot, we

More information

Discussion 02 Solutions

Discussion 02 Solutions STAT 400 Discussio 0 Solutios Spig 08. ~.5 ~.6 At the begiig of a cetai study of a goup of pesos, 5% wee classified as heavy smoes, 30% as light smoes, ad 55% as osmoes. I the fiveyea study, it was detemied

More information

( ) ( ) ( ) ( ) Solved Examples. JEE Main/Boards = The total number of terms in the expansion are 8.

( ) ( ) ( ) ( ) 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 information

Consider unordered sample of size r. This sample can be used to make r! Ordered samples (r! permutations). unordered sample

Consider 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 information

SEQUENCES AND SERIES

SEQUENCES AND SERIES Sequeces ad 6 Sequeces Ad SEQUENCES AND SERIES Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives rise to what is called a sequece. Sequeces

More information

Zeros of Polynomials

Zeros of Polynomials Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree

More information

Section 6.4: Series. Section 6.4 Series 413

Section 6.4: Series. Section 6.4 Series 413 ectio 64 eries 4 ectio 64: eries A couple decides to start a college fud for their daughter They pla to ivest $50 i the fud each moth The fud pays 6% aual iterest, compouded mothly How much moey will they

More information

On composite conformal mapping of an annulus to a plane with two holes

On composite conformal mapping of an annulus to a plane with two holes O composite cofomal mappig of a aulus to a plae with two holes Mila Batista (July 07) Abstact I the aticle we coside the composite cofomal map which maps aulus to ifiite egio with symmetic hole ad ealy

More information

IDENTITIES FOR THE NUMBER OF STANDARD YOUNG TABLEAUX IN SOME (k, l)-hooks

IDENTITIES FOR THE NUMBER OF STANDARD YOUNG TABLEAUX IN SOME (k, l)-hooks Sémiaie Lothaigie de Combiatoie 63 (010), Aticle B63c IDENTITIES FOR THE NUMBER OF STANDARD YOUNG TABLEAUX IN SOME (k, l)-hooks A. REGEV Abstact. Closed fomulas ae kow fo S(k,0;), the umbe of stadad Youg

More information

Lecture 24: Observability and Constructibility

Lecture 24: Observability and Constructibility ectue 24: Obsevability ad Costuctibility 7 Obsevability ad Costuctibility Motivatio: State feedback laws deped o a kowledge of the cuet state. I some systems, xt () ca be measued diectly, e.g., positio

More information

Lecture 2: Stress. 1. Forces Surface Forces and Body Forces

Lecture 2: Stress. 1. Forces Surface Forces and Body Forces Lectue : Stess Geophysicists study pheomea such as seismicity, plate tectoics, ad the slow flow of ocks ad mieals called ceep. Oe way they study these pheomea is by ivestigatig the defomatio ad flow of

More information

Infinite Sequences and Series

Infinite Sequences and Series Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet

More information

The number of r element subsets of a set with n r elements

The 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 information

EXAMPLES. Leader in CBSE Coaching. Solutions of BINOMIAL THEOREM A.V.T.E. by AVTE (avte.in) Class XI

EXAMPLES. 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 information

11.1 Arithmetic Sequences and Series

11.1 Arithmetic Sequences and Series 11.1 Arithmetic Sequeces ad Series A itroductio 1, 4, 7, 10, 13 9, 1, 7, 15 6., 6.6, 7, 7.4 ππ+, 3, π+ 6 Arithmetic Sequeces ADD To get ext term 35 1 7. 3π + 9, 4, 8, 16, 3 9, 3, 1, 1/ 3 1,1/ 4,1/16,1/

More information

Lesson 10: Limits and Continuity

Lesson 10: Limits and Continuity www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals

More information

CHAPTER 10 INFINITE SEQUENCES AND SERIES

CHAPTER 10 INFINITE SEQUENCES AND SERIES CHAPTER 10 INFINITE SEQUENCES AND SERIES 10.1 Sequeces 10.2 Ifiite Series 10.3 The Itegral Tests 10.4 Compariso Tests 10.5 The Ratio ad Root Tests 10.6 Alteratig Series: Absolute ad Coditioal Covergece

More information

Math 113 Exam 3 Practice

Math 113 Exam 3 Practice Math Exam Practice Exam will cover.-.9. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for you

More information

Algebra. Substitution in algebra. 3 Find the value of the following expressions if u = 4, k = 7 and t = 9.

Algebra. Substitution in algebra. 3 Find the value of the following expressions if u = 4, k = 7 and t = 9. lgeba Substitution in algeba Remembe... In an algebaic expession, lettes ae used as substitutes fo numbes. Example Find the value of the following expessions if s =. a) s + + = = s + + = = Example Find

More information

Section 1 of Unit 03 (Pure Mathematics 3) Algebra

Section 1 of Unit 03 (Pure Mathematics 3) Algebra Sectio 1 of Uit 0 (Pure Mathematics ) Algebra Recommeded Prior Kowledge Studets should have studied the algebraic techiques i Pure Mathematics 1. Cotet This Sectio should be studied early i the course

More information

EDEXCEL NATIONAL CERTIFICATE UNIT 4 MATHEMATICS FOR TECHNICIANS OUTCOME 4 - CALCULUS

EDEXCEL NATIONAL CERTIFICATE UNIT 4 MATHEMATICS FOR TECHNICIANS OUTCOME 4 - CALCULUS EDEXCEL NATIONAL CERTIFICATE UNIT 4 MATHEMATICS FOR TECHNICIANS OUTCOME 4 - CALCULUS TUTORIAL 1 - DIFFERENTIATION Use the elemetary rules of calculus arithmetic to solve problems that ivolve differetiatio

More information

Chapter 6 Overview: Sequences and Numerical Series. For the purposes of AP, this topic is broken into four basic subtopics:

Chapter 6 Overview: Sequences and Numerical Series. For the purposes of AP, this topic is broken into four basic subtopics: Chapter 6 Overview: Sequeces ad Numerical Series I most texts, the topic of sequeces ad series appears, at first, to be a side topic. There are almost o derivatives or itegrals (which is what most studets

More information

Appendix F: Complex Numbers

Appendix F: Complex Numbers Appedix F Complex Numbers F1 Appedix F: Complex Numbers Use the imagiary uit i to write complex umbers, ad to add, subtract, ad multiply complex umbers. Fid complex solutios of quadratic equatios. Write

More information

INFINITE SEQUENCES AND SERIES

INFINITE SEQUENCES AND SERIES 11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES 11.4 The Compariso Tests I this sectio, we will lear: How to fid the value of a series by comparig it with a kow series. COMPARISON TESTS

More information

Eton Education Centre JC 1 (2010) Consolidation quiz on Normal distribution By Wee WS (wenshih.wordpress.com) [ For SAJC group of students ]

Eton Education Centre JC 1 (2010) Consolidation quiz on Normal distribution By Wee WS (wenshih.wordpress.com) [ For SAJC group of students ] JC (00) Cosolidatio quiz o Normal distributio By Wee WS (weshih.wordpress.com) [ For SAJC group of studets ] Sped miutes o this questio. Q [ TJC 0/JC ] Mr Fruiti is the ower of a fruit stall sellig a variety

More information

Chapter 7: Numerical Series

Chapter 7: Numerical Series Chapter 7: Numerical Series Chapter 7 Overview: Sequeces ad Numerical Series I most texts, the topic of sequeces ad series appears, at first, to be a side topic. There are almost o derivatives or itegrals

More information

FIXED POINT AND HYERS-ULAM-RASSIAS STABILITY OF A QUADRATIC FUNCTIONAL EQUATION IN BANACH SPACES

FIXED POINT AND HYERS-ULAM-RASSIAS STABILITY OF A QUADRATIC FUNCTIONAL EQUATION IN BANACH SPACES IJRRAS 6 () July 0 www.apapess.com/volumes/vol6issue/ijrras_6.pdf FIXED POINT AND HYERS-UAM-RASSIAS STABIITY OF A QUADRATIC FUNCTIONA EQUATION IN BANACH SPACES E. Movahedia Behbaha Khatam Al-Abia Uivesity

More information

Solution to HW 3, Ma 1a Fall 2016

Solution to HW 3, Ma 1a Fall 2016 Solution to HW 3, Ma a Fall 206 Section 2. Execise 2: Let C be a subset of the eal numbes consisting of those eal numbes x having the popety that evey digit in the decimal expansion of x is, 3, 5, o 7.

More information

Minimization of the quadratic test function

Minimization of the quadratic test function Miimizatio of the quadatic test fuctio A quadatic fom is a scala quadatic fuctio of a vecto with the fom f ( ) A b c with b R A R whee A is assumed to be SPD ad c is a scala costat Note: A symmetic mati

More information

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 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 information

Linear Regression Demystified

Linear Regression Demystified Liear Regressio Demystified Liear regressio is a importat subject i statistics. I elemetary statistics courses, formulae related to liear regressio are ofte stated without derivatio. This ote iteds to

More information

Lecture 3 : Concentration and Correlation

Lecture 3 : Concentration and Correlation Lectue 3 : Cocetatio ad Coelatio 1. Talagad s iequality 2. Covegece i distibutio 3. Coelatio iequalities 1. Talagad s iequality Cetifiable fuctios Let g : R N be a fuctio. The a fuctio f : 1 2 Ω Ω L Ω

More information

A note on random minimum length spanning trees

A note on random minimum length spanning trees A ote o adom miimum legth spaig tees Ala Fieze Miklós Ruszikó Lubos Thoma Depatmet of Mathematical Scieces Caegie Mello Uivesity Pittsbugh PA15213, USA ala@adom.math.cmu.edu, usziko@luta.sztaki.hu, thoma@qwes.math.cmu.edu

More information

ON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS

ON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS Joual of Pue ad Alied Mathematics: Advaces ad Alicatios Volume 0 Numbe 03 Pages 5-58 ON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS ALI H HAKAMI Deatmet of Mathematics

More information

Chapter 6: Numerical Series

Chapter 6: Numerical Series Chapter 6: Numerical Series 327 Chapter 6 Overview: Sequeces ad Numerical Series I most texts, the topic of sequeces ad series appears, at first, to be a side topic. There are almost o derivatives or itegrals

More information

Sampling Error. Chapter 6 Student Lecture Notes 6-1. Business Statistics: A Decision-Making Approach, 6e. Chapter Goals

Sampling Error. Chapter 6 Student Lecture Notes 6-1. Business Statistics: A Decision-Making Approach, 6e. Chapter Goals Chapter 6 Studet Lecture Notes 6-1 Busiess Statistics: A Decisio-Makig Approach 6 th Editio Chapter 6 Itroductio to Samplig Distributios Chap 6-1 Chapter Goals After completig this chapter, you should

More information

Elementary Statistics

Elementary Statistics Elemetary Statistics M. Ghamsary, Ph.D. Sprig 004 Chap 0 Descriptive Statistics Raw Data: Whe data are collected i origial form, they are called raw data. The followig are the scores o the first test of

More information

Unit 6: Sequences and Series

Unit 6: Sequences and Series AMHS Hoors Algebra 2 - Uit 6 Uit 6: Sequeces ad Series 26 Sequeces Defiitio: A sequece is a ordered list of umbers ad is formally defied as a fuctio whose domai is the set of positive itegers. It is commo

More information

ELEMENTARY AND COMPOUND EVENTS PROBABILITY

ELEMENTARY AND COMPOUND EVENTS PROBABILITY Euopea Joual of Basic ad Applied Scieces Vol. 5 No., 08 ELEMENTARY AND COMPOUND EVENTS PROBABILITY William W.S. Che Depatmet of Statistics The Geoge Washigto Uivesity Washigto D.C. 003 E-mail: williamwsche@gmail.com

More information

Further Exploration of Patterns

Further Exploration of Patterns Further Exploratio of Patters Abstract Quadratic Patters a+b+c 4a+b+c 9a+3b+c 16a+4b+c 5a+5b+c 1 st chage 3a+b 5a+b 7a+b 9a+b d chage a a a 1 Coefficiet of is ( a) a. Costat secod chage, therefore it is

More information

BINOMIAL THEOREM & ITS SIMPLE APPLICATION

BINOMIAL 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 information

Optimization Methods: Linear Programming Applications Assignment Problem 1. Module 4 Lecture Notes 3. Assignment Problem

Optimization Methods: Linear Programming Applications Assignment Problem 1. Module 4 Lecture Notes 3. Assignment Problem Optimizatio Methods: Liear Programmig Applicatios Assigmet Problem Itroductio Module 4 Lecture Notes 3 Assigmet Problem I the previous lecture, we discussed about oe of the bech mark problems called trasportatio

More information

WORKING WITH NUMBERS

WORKING WITH NUMBERS 1 WORKING WITH NUMBERS WHAT YOU NEED TO KNOW The defiitio of the differet umber sets: is the set of atural umbers {0, 1,, 3, }. is the set of itegers {, 3,, 1, 0, 1,, 3, }; + is the set of positive itegers;

More information

Summary: Congruences. j=1. 1 Here we use the Mathematica syntax for the function. In Maple worksheets, the function

Summary: Congruences. j=1. 1 Here we use the Mathematica syntax for the function. In Maple worksheets, the function Summary: Cogrueces j whe divided by, ad determiig the additive order of a iteger mod. As described i the Prelab sectio, cogrueces ca be thought of i terms of coutig with rows, ad for some questios this

More information

When two numbers are written as the product of their prime factors, they are in factored form.

When two numbers are written as the product of their prime factors, they are in factored form. 10 1 Study Guide Pages 420 425 Factos Because 3 4 12, we say that 3 and 4 ae factos of 12. In othe wods, factos ae the numbes you multiply to get a poduct. Since 2 6 12, 2 and 6 ae also factos of 12. The

More information

3sin A 1 2sin B. 3π x is a solution. 1. If A and B are acute positive angles satisfying the equation 3sin A 2sin B 1 and 3sin 2A 2sin 2B 0, then A 2B

3sin A 1 2sin B. 3π x is a solution. 1. If A and B are acute positive angles satisfying the equation 3sin A 2sin B 1 and 3sin 2A 2sin 2B 0, then A 2B 1. If A ad B are acute positive agles satisfyig the equatio 3si A si B 1 ad 3si A si B 0, the A B (a) (b) (c) (d) 6. 3 si A + si B = 1 3si A 1 si B 3 si A = cosb Also 3 si A si B = 0 si B = 3 si A Now,

More information