XT - MATHS Grade 12. Date: 2010/06/29. Subject: Series and Sequences 1: Arithmetic Total Marks: 84 = 2 = 2 1. FALSE 10.
|
|
- Nelson Melton
- 5 years ago
- Views:
Transcription
1 ubject: eries ad equeces 1: Arithmetic otal Mars: 8 X - MAH Grade 1 Date: 010/0/ 1. FALE 10 Explaatio: his series is arithmetic as d 1 ad d 15 1 he sum of a arithmetic series is give by [ a ( ] a represets the first term of this series: a represets the umber of terms i this series: 10? d represets the costat differece betwee the terms of this series: d he [ () ( 1)() ] [ ] [ 0 ] he sum of this series must be larger tha 00. herefore [ 0 ] > 00 [ 10 ] > > > 0 ( 10)( 0) > 0 As the umber of terms caot be egative or a fractio, > 10, i.e. oly eleve or more terms will give a sum of larger tha 00.. FALE Explaatio: he first ad secod terms of a arithmetic progressio are 1 ad 1. herefore d he th term of a arithmetic progressio: a ( a is the first term: a 1 d is the costat differece: d is the term umber: the 5 th term must be foud, therefore 5 herefore Page 1
2 1 (5 1)() 5 1 ( ) he thirty-fifth term is therefore 17 ad ot 1.. B Explaatio: First determie if this is a arithmetic or a geometric series: d d r 1 r 1 1 here is a costat differece betwee successive terms ad ot a costat ratio, therefore this series is arithmetic. he sum formula for a arithmetic progressio: a ( a represets the first term of this series: a l represets the last term of this series: l 5 [ ] OR [ a l ] represets the umber of terms i this series:? d represets the costat differece betwee the terms of this series: d 5 represets the last term of this series: 5 is the sum of terms of this series:? he umber of terms must be determied before the sum ca be calculated. he th term of aarithmetic progressio: a ( 5 ( 1)(5) Now [ () ( 1)(5) ] [ 8 50 ] 58 5 OR [ ] C 8 Explaatio: he multiples are: ; 7 ; 0 ; ; 7 ; 10 his is a arithmetic progressio as is added to each term to get the ext term. o fid the umber of terms to be added, you eed to use the -formula of a AP: Page
3 a ( 10 ( 1)() 10 Now ( 1) [ a d ] () ( 1)() 8 [ ] [ ] 7 7 OR [ l ] [ 10 ] a D Explaatio: o determie whether a sequece is arithmetic, you eed to fid the differece (d) betwee cosecutive terms. A: d 1 or d or d he aswers are the same, therefore sequece A is arithmetic. B: d 1 or d 7 18 he aswers are ot the same, therefore sequece B is ot arithmetic. C: d 1 or d (x ) ( x ) (5x ) (x ) x x 5x x x 1 x 1 he aswers are the same, therefore sequece C is arithmetic Explaatio: 1 ( 1) [(1) 1] [() 1] [() 1] he terms of this series differ with, this series is therefore arithmetic. he sum formula for a arithmetic progressio: Page
4 a represets the first term of this series: a 1 l represets the last term of this series: l - 1 [ a ( ] OR [ a l ] represets the umber of terms i this series:? d represets the costat differece betwee the terms of this series: d represets the last term of this series: - 1 is the sum of terms of this series: Now [ a ( ] [ (1) ( 1)() ] [ ] OR [ a l ] [ 1 ( 1) ] [ ] ± 1 ± 1 As represets the umber of terms, - 1 is a ivalid solutio Explaatio: he -formula for a arithmetic progressio: a ( a is the first term: a d is the costat differece: d -5 is the umber of terms to be added: 1 herefore 1 () (1 1)( 5) 1 1 (18)( 5) ( 8) 1 ( ) 8. 0,5 Explaatio: Arithmetic Mea , 5 Page
5 . (1) 1,,, 7,, 18 () is ot () Lucas Explaatio: 1. his is a secod order differece equatio, where the first two terms are give. 1, he differece betwee the terms is ot costat. herefor this is ot a arithmetic patter.. Eduoardo Lucas discovered this umber patter, whe he compared differet startig poits to the better ow Fiboacci series. How ice to have a umber patter amed after you! 10. (1) 7, 10 () quadratic () - 1, 1 - Explaatio: 1. tudy the patter: -,, 1, 0, Now determie the differeces: Determie the differeces of the differeces: o determie the ext two terms, this patter of differeces must be maitaied. Hece the sixth ad seveth terms must be ad 8 larger tha its predecessors respectively ad A liear patter has its first differece costat. A quadratic patter has its secod differece costat. A cubic patter has its third differece costat.. he formula - 10 geerates the patter, but it is a explicit formula, ie ot a recursive formula. he formula 1, 1 - geerates 1-1 () - 8 () 1 1 () (5) his is the patter required.. he formula 1 1, 1 - geerates: (-) () his is ot the patter required. Page 5
6 . (1) a 1d 1 () 5 () - Explaatio: ubstitute the two sets of give iformatio ito the -formula for a arithmetic progressio: a ( th term is : a ( 1 a 5d... [1] th 1 term is 1 : a (1 1 1 a 1d... [] ubtract [1] from []: 1 a 5d 1 a 1d... []... [1] 5 7d 5 d a represets the first term of the sequece. o fid the value of a, substitute d 5 ito -1 a 5d: 1 a 5( 5) 1 a 5 a a Explaatio: he th term of a arithmetic sequece: a ( a is the first term of the sequece, therefore a -7. d is the differece betwee the terms of the sequece: d or 1 ( ) ( 7) 7 d ( 1) ( ) 1 herefore: 7 ( 1)() [ remais as it is uow ] Explaatio: t 1 (t ) [(1) ] [() ] [() ]... [( ) ] ( ) Page
7 his is a arithmetic series with d. he sum formula for a AP: [ a ( ] a represets the first term of this series: a 1 represets the umber of terms i this series: d represets the costat differece betwee the terms of this series: d is the sum of terms of this series: < 58 Now [ ( 1) ( 1)() ] [ ] [ ] herefore < < 0 ( )( ) < 0 As represets the term umber of the last term, oly 1; ; ; ; ca be valid values for ( < < ). herefore, the greatest value that could have is. 1. A Explaatio: herefore : ( 8) B 5 Explaatio: I this arithmetic sequece: a, d?, ad he... a ( ( 0 d 0 d herefore: Page 7
8 x ad y 1 d 0 d 0 15 Questios, 8 Pages Page 8
MISCELLANEOUS SEQUENCES & SERIES QUESTIONS
MISCELLANEOUS SEQUENCES & SERIES QUESTIONS Questio (***+) Evaluate the followig sum 30 r ( 2) 4r 78. 3 MP2-V, 75,822,200 Questio 2 (***+) Three umbers, A, B, C i that order, are i geometric progressio
More informationSection 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 informationSEQUENCES 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 informationSEQUENCES 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 informationFind a formula for the exponential function whose graph is given , 1 2,16 1, 6
Math 4 Activity (Due by EOC Apr. ) Graph the followig epoetial fuctios by modifyig the graph of f. Fid the rage of each fuctio.. g. g. g 4. g. g 6. g Fid a formula for the epoetial fuctio whose graph is
More informationVERY 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 informationARITHMETIC PROGRESSION
CHAPTER 5 ARITHMETIC PROGRESSION Poits to Remember :. A sequece is a arragemet of umbers or objects i a defiite order.. A sequece a, a, a 3,..., a,... is called a Arithmetic Progressio (A.P) if there exists
More informationInfinite 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 informationSeptember 2012 C1 Note. C1 Notes (Edexcel) Copyright - For AS, A2 notes and IGCSE / GCSE worksheets 1
September 0 s (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets September 0 Copyright
More informationMath 2412 Review 3(answers) kt
Math 4 Review 3(aswers) kt A t A e. If the half-life of radium is 690 years, ad you have 0 grams ow, how much will be preset i 50 years (rouded to three decimal places)?. The decay of radium is modeled
More informationUnit 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 informationSEQUENCE 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 information07 - SEQUENCES AND SERIES Page 1 ( Answers at he end of all questions ) b, z = n
07 - SEQUENCES AND SERIES Page ( Aswers at he ed of all questios ) ( ) If = a, y = b, z = c, where a, b, c are i A.P. ad = 0 = 0 = 0 l a l
More informationCHAPTER - 9 SEQUENCES AND SERIES KEY POINTS A sequece is a fuctio whose domai is the set N of atural umbers. A sequece whose rage is a subset of R is called a real sequece. Geeral A.P. is, a, a + d, a
More informationARITHMETIC PROGRESSIONS
CHAPTER 5 ARITHMETIC PROGRESSIONS (A) Mai Cocepts ad Results A arithmetic progressio (AP) is a list of umbers i which each term is obtaied by addig a fixed umber d to the precedig term, except the first
More informationObjective Mathematics
. If sum of '' terms of a sequece is give by S Tr ( )( ), the 4 5 67 r (d) 4 9 r is equal to : T. Let a, b, c be distict o-zero real umbers such that a, b, c are i harmoic progressio ad a, b, c are i arithmetic
More informationTopic 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 informationSequences, Sums, and Products
CSCE 222 Discrete Structures for Computig Sequeces, Sums, ad Products Dr. Philip C. Ritchey Sequeces A sequece is a fuctio from a subset of the itegers to a set S. A discrete structure used to represet
More informationChapter 6. Progressions
Chapter 6 Progressios Evidece is foud that Babyloias some 400 years ago, kew of arithmetic ad geometric progressios. Amog the Idia mathematicias, Aryabhata (470 AD) was the first to give formula for the
More informationProgressions. ILLUSTRATION 1 11, 7, 3, -1, i s an A.P. whose first term is 11 and the common difference 7-11=-4.
Progressios SEQUENCE A sequece is a fuctio whose domai is the set N of atural umbers. REAL SEQUENCE A Sequece whose rage is a subset of R is called a real sequece. I other words, a real sequece is a fuctio
More informationEssential Question How can you recognize an arithmetic sequence from its graph?
. Aalyzig Arithmetic Sequeces ad Series COMMON CORE Learig Stadards HSF-IF.A.3 HSF-BF.A. HSF-LE.A. Essetial Questio How ca you recogize a arithmetic sequece from its graph? I a arithmetic sequece, the
More information3sin 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 informationis 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 informationSection 7 Fundamentals of Sequences and Series
ectio Fudametals of equeces ad eries. Defiitio ad examples of sequeces A sequece ca be thought of as a ifiite list of umbers. 0, -, -0, -, -0...,,,,,,. (iii),,,,... Defiitio: A sequece is a fuctio which
More informationStarting with a time of 1 second, the partial sums of the time series form the sequence 1, 3 2, 7 4, 1 5
- OBJECTIVE Determie whether a series is coverget or diverget. Coverget ad Diverget Series HISTORY The Greek philosopher Zeo of Elea (c. 90 30 B.C.) proposed several perplexig riddles, or paradoxes. Oe
More informationChapter 10: Power Series
Chapter : Power Series 57 Chapter Overview: Power Series The reaso series are part of a Calculus course is that there are fuctios which caot be itegrated. All power series, though, ca be itegrated because
More informationFurther 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 informationCHAPTER 1 SEQUENCES AND INFINITE SERIES
CHAPTER SEQUENCES AND INFINITE SERIES SEQUENCES AND INFINITE SERIES (0 meetigs) Sequeces ad limit of a sequece Mootoic ad bouded sequece Ifiite series of costat terms Ifiite series of positive terms Alteratig
More informationChapter Vectors
Chapter 4. Vectors fter readig this chapter you should be able to:. defie a vector. add ad subtract vectors. fid liear combiatios of vectors ad their relatioship to a set of equatios 4. explai what it
More informationChapter 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 informationCreated by T. Madas SERIES. Created by T. Madas
SERIES SUMMATIONS BY STANDARD RESULTS Questio (**) Use stadard results o summatios to fid the value of 48 ( r )( 3r ). 36 FP-B, 66638 Questio (**+) Fid, i fully simplified factorized form, a expressio
More informationQ.11 If S be the sum, P the product & R the sum of the reciprocals of a GP, find the value of
Brai Teasures Progressio ad Series By Abhijit kumar Jha EXERCISE I Q If the 0th term of a HP is & st term of the same HP is 0, the fid the 0 th term Q ( ) Show that l (4 36 08 up to terms) = l + l 3 Q3
More informationChapter 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 informationMIXED REVIEW of Problem Solving
MIXED REVIEW of Problem Solvig STATE TEST PRACTICE classzoe.com Lessos 2.4 2.. MULTI-STEP PROBLEM A ball is dropped from a height of 2 feet. Each time the ball hits the groud, it bouces to 70% of its previous
More informationChapter 9 Sequences, Series, and Probability Section 9.4 Mathematical Induction
Chapter 9 equeces, eries, ad Probability ectio 9. Mathematical Iductio ectio Objectives: tudets will lear how to use mathematical iductio to prove statemets ivolvig a positive iteger, recogize patters
More informationWORKING 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 informationMath 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 information2.4 - Sequences and Series
2.4 - Sequeces ad Series Sequeces A sequece is a ordered list of elemets. Defiitio 1 A sequece is a fuctio from a subset of the set of itegers (usually either the set 80, 1, 2, 3,... < or the set 81, 2,
More informationCALCULATION OF FIBONACCI VECTORS
CALCULATION OF FIBONACCI VECTORS Stuart D. Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithaca.edu ad Dai Novak Departmet of Mathematics, Ithaca College
More informationDifferent kinds of Mathematical Induction
Differet ids of Mathematical Iductio () Mathematical Iductio Give A N, [ A (a A a A)] A N () (First) Priciple of Mathematical Iductio Let P() be a propositio (ope setece), if we put A { : N p() is true}
More informationRevision Topic 1: Number and algebra
Revisio Topic : Number ad algebra Chapter : Number Differet types of umbers You eed to kow that there are differet types of umbers ad recogise which group a particular umber belogs to: Type of umber Symbol
More informationsin(n) + 2 cos(2n) n 3/2 3 sin(n) 2cos(2n) n 3/2 a n =
60. Ratio ad root tests 60.1. Absolutely coverget series. Defiitio 13. (Absolute covergece) A series a is called absolutely coverget if the series of absolute values a is coverget. The absolute covergece
More informationADDITIONAL MATHEMATICS FORM 5 MODULE 2
PROGRAM DIDIK CEMERLANG AKADEMIK SPM ADDITIONAL MATHEMATICS FORM 5 MODULE 2 PROGRESSIONS (Geometric Progressio) ORGANISED BY: JABATAN PELAJARAN NEGERI PULAU PINANG CHAPTER 2 : GEOMETRIC PROGRESSIONS Cotets
More information1 Generating functions for balls in boxes
Math 566 Fall 05 Some otes o geeratig fuctios Give a sequece a 0, a, a,..., a,..., a geeratig fuctio some way of represetig the sequece as a fuctio. There are may ways to do this, with the most commo ways
More information+ au n+1 + bu n = 0.)
Lecture 6 Recurreces - kth order: u +k + a u +k +... a k u k 0 where a... a k are give costats, u 0... u k are startig coditios. (Simple case: u + au + + bu 0.) How to solve explicitly - first, write characteristic
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More information6.3 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 informationSequences. A Sequence is a list of numbers written in order.
Sequeces A Sequece is a list of umbers writte i order. {a, a 2, a 3,... } The sequece may be ifiite. The th term of the sequece is the th umber o the list. O the list above a = st term, a 2 = 2 d term,
More informationYou may work in pairs or purely individually for this assignment.
CS 04 Problem Solvig i Computer Sciece OOC Assigmet 6: Recurreces You may work i pairs or purely idividually for this assigmet. Prepare your aswers to the followig questios i a plai ASCII text file or
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 informationLecture 3. Digital Signal Processing. Chapter 3. z-transforms. Mikael Swartling Nedelko Grbic Bengt Mandersson. rev. 2016
Lecture 3 Digital Sigal Processig Chapter 3 z-trasforms Mikael Swartlig Nedelko Grbic Begt Madersso rev. 06 Departmet of Electrical ad Iformatio Techology Lud Uiversity z-trasforms We defie the z-trasform
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More information3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials
Math 60 www.timetodare.com 3. Properties of Divisio 3.3 Zeros of Polyomials 3.4 Complex ad Ratioal Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered
More informationSeries: Infinite Sums
Series: Ifiite Sums Series are a way to mae sese of certai types of ifiitely log sums. We will eed to be able to do this if we are to attai our goal of approximatig trascedetal fuctios by usig ifiite degree
More informationAddition: Property Name Property Description Examples. a+b = b+a. a+(b+c) = (a+b)+c
Notes for March 31 Fields: A field is a set of umbers with two (biary) operatios (usually called additio [+] ad multiplicatio [ ]) such that the followig properties hold: Additio: Name Descriptio Commutativity
More informationSeries III. Chapter Alternating Series
Chapter 9 Series III With the exceptio of the Null Sequece Test, all the tests for series covergece ad divergece that we have cosidered so far have dealt oly with series of oegative terms. Series with
More informationChapter 4. Fourier Series
Chapter 4. Fourier Series At this poit we are ready to ow cosider the caoical equatios. Cosider, for eample the heat equatio u t = u, < (4.) subject to u(, ) = si, u(, t) = u(, t) =. (4.) Here,
More informationComplex Numbers Solutions
Complex Numbers Solutios Joseph Zoller February 7, 06 Solutios. (009 AIME I Problem ) There is a complex umber with imagiary part 64 ad a positive iteger such that Fid. [Solutio: 697] 4i + + 4i. 4i 4i
More information6.1. Sequences as Discrete Functions. Investigate
6.1 Sequeces as Discrete Fuctios The word sequece is used i everyday laguage. I a sequece, the order i which evets occur is importat. For example, builders must complete work i the proper sequece to costruct
More information10.2 Infinite Series Contemporary Calculus 1
10. Ifiite Series Cotemporary Calculus 1 10. INFINITE SERIES Our goal i this sectio is to add together the umbers i a sequece. Sice it would take a very log time to add together the ifiite umber of umbers,
More informationUNIVERSITY OF NORTHERN COLORADO MATHEMATICS CONTEST. First Round For all Colorado Students Grades 7-12 November 3, 2007
UNIVERSITY OF NORTHERN COLORADO MATHEMATICS CONTEST First Roud For all Colorado Studets Grades 7- November, 7 The positive itegers are,,, 4, 5, 6, 7, 8, 9,,,,. The Pythagorea Theorem says that a + b =
More informationf(x) dx as we do. 2x dx x also diverges. Solution: We compute 2x dx lim
Math 3, Sectio 2. (25 poits) Why we defie f(x) dx as we do. (a) Show that the improper itegral diverges. Hece the improper itegral x 2 + x 2 + b also diverges. Solutio: We compute x 2 + = lim b x 2 + =
More informationOn Some Properties of Digital Roots
Advaces i Pure Mathematics, 04, 4, 95-30 Published Olie Jue 04 i SciRes. http://www.scirp.org/joural/apm http://dx.doi.org/0.436/apm.04.46039 O Some Properties of Digital Roots Ilha M. Izmirli Departmet
More informationMATH 304: MIDTERM EXAM SOLUTIONS
MATH 304: MIDTERM EXAM SOLUTIONS [The problems are each worth five poits, except for problem 8, which is worth 8 poits. Thus there are 43 possible poits.] 1. Use the Euclidea algorithm to fid the greatest
More informationFINALTERM EXAMINATION Fall 9 Calculus & Aalytical Geometry-I Questio No: ( Mars: ) - Please choose oe Let f ( x) is a fuctio such that as x approaches a real umber a, either from left or right-had-side,
More informationdifference equations introduction Generating the terms of a sequence defi ned by a fi rst order difference equation
6 differece equatios areas of STudY Geeratio of the terms of a sequece from a differece equatio, graphical represetatio of such a sequece ad iterpretatio of the graph of the sequece Arithmetic ad geometric
More informationUNIT #5. Lesson #2 Arithmetic and Geometric Sequences. Lesson #3 Summation Notation. Lesson #4 Arithmetic Series. Lesson #5 Geometric Series
UNIT #5 SEQUENCES AND SERIES Lesso # Sequeces Lesso # Arithmetic ad Geometric Sequeces Lesso #3 Summatio Notatio Lesso #4 Arithmetic Series Lesso #5 Geometric Series Lesso #6 Mortgage Paymets COMMON CORE
More informationENGI Series Page 6-01
ENGI 3425 6 Series Page 6-01 6. Series Cotets: 6.01 Sequeces; geeral term, limits, covergece 6.02 Series; summatio otatio, covergece, divergece test 6.03 Stadard Series; telescopig series, geometric series,
More informationEXERCISE - 01 CHECK YOUR GRASP
J-Mathematics XRCIS - 0 CHCK YOUR GRASP SLCT TH CORRCT ALTRNATIV (ONLY ON CORRCT ANSWR). The maximum value of the sum of the A.P. 0, 8, 6,,... is - 68 60 6. Let T r be the r th term of a A.P. for r =,,,...
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science. BACKGROUND EXAM September 30, 2004.
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Departmet of Electrical Egieerig ad Computer Sciece 6.34 Discrete Time Sigal Processig Fall 24 BACKGROUND EXAM September 3, 24. Full Name: Note: This exam is closed
More information6.003 Homework #3 Solutions
6.00 Homework # Solutios Problems. Complex umbers a. Evaluate the real ad imagiary parts of j j. π/ Real part = Imagiary part = 0 e Euler s formula says that j = e jπ/, so jπ/ j π/ j j = e = e. Thus the
More informationNUMERICAL METHODS FOR SOLVING EQUATIONS
Mathematics Revisio Guides Numerical Methods for Solvig Equatios Page 1 of 11 M.K. HOME TUITION Mathematics Revisio Guides Level: GCSE Higher Tier NUMERICAL METHODS FOR SOLVING EQUATIONS Versio:. Date:
More informationProperties and Tests of Zeros of Polynomial Functions
Properties ad Tests of Zeros of Polyomial Fuctios The Remaider ad Factor Theorems: Sythetic divisio ca be used to fid the values of polyomials i a sometimes easier way tha substitutio. This is show by
More informationT1.1 Lesson 3 - Arithmetic & Geometric Series & Summation Notation
Fast Five T. Lesso 3 - Arithmetic & Geometric eries & ummatio Notatio Math L - atowski Fid the sum of the first 00 umbers Outlie a way to solve this problem ad the carry out your pla Fid the sum of the
More informationCHAPTER 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 informationINFINITE 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 informationEton 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 informationAn alternating series is a series where the signs alternate. Generally (but not always) there is a factor of the form ( 1) n + 1
Calculus II - Problem Solvig Drill 20: Alteratig Series, Ratio ad Root Tests Questio No. of 0 Istructios: () Read the problem ad aswer choices carefully (2) Work the problems o paper as eeded (3) Pick
More informationNATIONAL JUNIOR COLLEGE SENIOR HIGH 1 PROMOTIONAL EXAMINATIONS Higher 2
NATIONAL JUNIOR COLLEGE SENIOR HIGH PROMOTIONAL EXAMINATIONS Higher MATHEMATICS 9758 9 September 06 hours Additioal Materials: Aswer Paper List of Formulae (MF6) Cover Sheet READ THESE INSTRUCTIONS FIRST
More informationRoberto s Notes on Infinite Series Chapter 1: Sequences and series Section 3. Geometric series
Roberto s Notes o Ifiite Series Chapter 1: Sequeces ad series Sectio Geometric series What you eed to kow already: What a ifiite series is. The divergece test. What you ca le here: Everythig there is to
More informationMATH301 Real Analysis (2008 Fall) Tutorial Note #7. k=1 f k (x) converges pointwise to S(x) on E if and
MATH01 Real Aalysis (2008 Fall) Tutorial Note #7 Sequece ad Series of fuctio 1: Poitwise Covergece ad Uiform Covergece Part I: Poitwise Covergece Defiitio of poitwise covergece: A sequece of fuctios f
More informationSubstitute these values into the first equation to get ( z + 6) + ( z + 3) + z = 27. Then solve to get
Problem ) The sum of three umbers is 7. The largest mius the smallest is 6. The secod largest mius the smallest is. What are the three umbers? [Problem submitted by Vi Lee, LCC Professor of Mathematics.
More informationMath 475, Problem Set #12: Answers
Math 475, Problem Set #12: Aswers A. Chapter 8, problem 12, parts (b) ad (d). (b) S # (, 2) = 2 2, sice, from amog the 2 ways of puttig elemets ito 2 distiguishable boxes, exactly 2 of them result i oe
More informationMath 113 Exam 3 Practice
Math Exam Practice Exam 4 will cover.-., 0. ad 0.. Note that eve though. was tested i exam, questios from that sectios may also be o this exam. For practice problems o., refer to the last review. This
More informationRecurrence Relations
Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial(-)); } Let t be the umber of multiplicatios eeded to calculate factorial(). The
More informationSection A assesses the Units Numerical Analysis 1 and 2 Section B assesses the Unit Mathematics for Applied Mathematics
X0/70 NATIONAL QUALIFICATIONS 005 MONDAY, MAY.00 PM 4.00 PM APPLIED MATHEMATICS ADVANCED HIGHER Numerical Aalysis Read carefully. Calculators may be used i this paper.. Cadidates should aswer all questios.
More informationFriday 20 May 2016 Morning
Oxford Cambridge ad RSA Friday 0 May 06 Morig AS GCE MATHEMATICS (MEI) 4755/0 Further Cocepts for Advaced Mathematics (FP) QUESTION PAPER * 6 8 6 6 9 5 4 * Cadidates aswer o the Prited Aswer Boo. OCR supplied
More information11.1 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 informationThe picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled
1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how
More informationZeros 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 informationSums, products and sequences
Sums, products ad sequeces How to write log sums, e.g., 1+2+ (-1)+ cocisely? i=1 Sum otatio ( sum from 1 to ): i 3 = 1 + 2 + + If =3, i=1 i = 1+2+3=6. The ame ii does ot matter. Could use aother letter
More informationCalculus Sequences and Series FAMAT State Convention For all questions, answer E. NOTA means none of the above answers are correct.
Calculus Sequeces ad Series FAMAT State Covetio 005 For all questios, aswer meas oe of the above aswers are correct.. What should be the et logical term i the sequece, 5, 9,, 0? A. 6 B. 7 8 9. Give the
More informationMETRO EAST EDUCATION DISTRICT NATIONAL SENIOR CERTIFICATE GRADE 12 MATHEMATICS PAPER 1 SEPTEMBER 2014
METRO EAST EDUCATION DISTRICT NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS PAPER SEPTEMBER 04 MARKS: 50 TIME: 3 hours This paper cosists of 7 pages ad a iformatio sheet. GR Mathematics- P MEED September
More informationSequences, Mathematical Induction, and Recursion. CSE 2353 Discrete Computational Structures Spring 2018
CSE 353 Discrete Computatioal Structures Sprig 08 Sequeces, Mathematical Iductio, ad Recursio (Chapter 5, Epp) Note: some course slides adopted from publisher-provided material Overview May mathematical
More informationPhysics 116A Solutions to Homework Set #1 Winter Boas, problem Use equation 1.8 to find a fraction describing
Physics 6A Solutios to Homework Set # Witer 0. Boas, problem. 8 Use equatio.8 to fid a fractio describig 0.694444444... Start with the formula S = a, ad otice that we ca remove ay umber of r fiite decimals
More informationPolynomials with Rational Roots that Differ by a Non-zero Constant. Generalities
Polyomials with Ratioal Roots that Differ by a No-zero Costat Philip Gibbs The problem of fidig two polyomials P(x) ad Q(x) of a give degree i a sigle variable x that have all ratioal roots ad differ by
More informationMATH 31B: MIDTERM 2 REVIEW
MATH 3B: MIDTERM REVIEW JOE HUGHES. Evaluate x (x ) (x 3).. Partial Fractios Solutio: The umerator has degree less tha the deomiator, so we ca use partial fractios. Write x (x ) (x 3) = A x + A (x ) +
More informationCSE 1400 Applied Discrete Mathematics Number Theory and Proofs
CSE 1400 Applied Discrete Mathematics Number Theory ad Proofs Departmet of Computer Scieces College of Egieerig Florida Tech Sprig 01 Problems for Number Theory Backgroud Number theory is the brach of
More informationGRADE 12 SEPTEMBER 2015 MATHEMATICS P1
NATIONAL SENIOR CERTIFICATE GRADE 1 SEPTEMBER 015 MATHEMATICS P1 MARKS: 150 TIME: 3 hours *MATHE1* This questio paper cosists of 10 pages, icludig a iformatio sheet. MATHEMATICS P1 (EC/SEPTEMBER 015) INSTRUCTIONS
More information6Sequences. 6.1 Kick off with CAS 6.2 Arithmetic sequences 6.3 Geometric sequences 6.4 Recurrence relations 6.5 Review
6Sequeces 6.1 Kick off with CAS 6.2 Arithmetic sequeces 6.3 Geometric sequeces 6.4 Recurrece relatios 6.5 Review 6.1 Kick off with CAS Explorig the Fiboacci sequece with CAS The Fiboacci sequece is a sequece
More information