MISCELLANEOUS SEQUENCES & SERIES QUESTIONS

Created 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

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

Section 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

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

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

Sequences, 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

XT - MATHS Grade 12. Date: 2010/06/29. Subject: Series and Sequences 1: Arithmetic Total Marks: 84 = 2 = 2 1. FALSE 10.

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

Objective 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

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

2.4 - 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,

CHAPTER - 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

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/

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

Progressions. 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

The 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

07 - 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

T1.1 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

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

Q.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

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

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

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

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,

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of

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

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

a n b. n a n Test Review for Lessons 9.1 and 9.3 Show all work on a separate sheet of paper for full credit.

Precalculus Test Review for Lessos. ad. Name: Date: Hour: Show all work o a separate sheet of paper for full credit.. Write the first four terms of the sequeces give by a. a b. a 7. Write the first five

Chapter 6 Infinite Series

Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat

Chapter 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

FINALTERM 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,

ARITHMETIC 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

ARITHMETIC 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

Math 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

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

ROSE WONG. f(1) f(n) where L the average value of f(n). In this paper, we will examine averages of several different arithmetic functions.

AVERAGE VALUES OF ARITHMETIC FUNCTIONS ROSE WONG Abstract. I this paper, we will preset problems ivolvig average values of arithmetic fuctios. The arithmetic fuctios we discuss are: (1)the umber of represetatios

Chapter 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

LINEAR RECURSION RELATIONS - LESSON FOUR SECOND-ORDER LINEAR RECURSION RELATIONS

LINEAR RECURSION RELATIONS - LESSON FOUR SECOND-ORDER LINEAR RECURSION RELATIONS BROTHER ALFRED BROUSSEAU St. Mary's College, Califoria Give a secod-order liear recursio relatio (.1) T. 1 = a T + b T 1,

SOLUTIONS TO PRISM PROBLEMS Junior Level 2014

SOLUTIONS TO PRISM PROBLEMS Juior Level 04. (B) Sice 50% of 50 is 50 5 ad 50% of 40 is the secod by 5 0 5. 40 0, the first exceeds. (A) Oe way of comparig the magitudes of the umbers,,, 5 ad 0.7 is 4 5

Find 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

Sequences and Series 4

Sequeces ad Series 4 LEARNING OBJECTIVES I this chapter, you lear how to use GC to Fid the th term of a Sequece. Fid the Sum of a Sequece. Fid the Sum to Ifiity of a GP. Solve Quadratic Equatio uder EQUA

Sums, 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

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

SEQUENCES AND SERIES

Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9. Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say that

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

CHAPTER 2 SEQUENCES AND SERIES OUTLINE Geometric Series Infinite Geometric Series and Sigma Notation 4 Review

CHAPTER SEQUENCES AND SERIES OUTLINE Day Sectio Topic.7 Geometric Sequeces.8 Geometric Series 3.9 Ifiite Geometric Series ad Sigma Notatio 4 Review 5 Review 6 Chapter Test Terry Fox Math 007 .7 GEOMETRIC

Quiz No. 1. ln n n. 1. Define: an infinite sequence A function whose domain is N 2. Define: a convergent sequence A sequence that has a limit

Quiz No.. Defie: a ifiite sequece A fuctio whose domai is N 2. Defie: a coverget sequece A sequece that has a limit 3. Is this sequece coverget? Why or why ot? l Yes, it is coverget sice L=0 by LHR. INFINITE

Essential 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

AP CALCULUS BC 2014 SCORING GUIDELINES

AP CALCULUS BC 04 SCORING GUIDELINES Questio 6 + The Taylor series for a fuctio f about x = is give by ( ) ( x ) ad coverges to f( x ) for = x < R, where R is the radius of covergece of the Taylor series.

Chapter 5.4 Practice Problems

EXPECTED SKILLS: Chapter 5.4 Practice Problems Uderstad ad kow how to evaluate the summatio (sigma) otatio. Be able to use the summatio operatio s basic properties ad formulas. (You do ot eed to memorize

ADDITIONAL 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

( a) ( ) 1 ( ) 2 ( ) ( ) 3 3 ( ) =!

.8,.9: Taylor ad Maclauri Series.8. Although we were able to fid power series represetatios for a limited group of fuctios i the previous sectio, it is ot immediately obvious whether ay give fuctio has

Sequences. 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,

Infinite Series and Improper Integrals

8 Special Fuctios Ifiite Series ad Improper Itegrals Ifiite series are importat i almost all areas of mathematics ad egieerig I additio to umerous other uses, they are used to defie certai fuctios ad to

ENGI 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,

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

4.1 Sigma Notation and Riemann Sums

0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas

We are mainly going to be concerned with power series in x, such as. (x)} converges - that is, lims N n

Review of Power Series, Power Series Solutios A power series i x - a is a ifiite series of the form c (x a) =c +c (x a)+(x a) +... We also call this a power series cetered at a. Ex. (x+) is cetered at

MATH4822E FOURIER ANALYSIS AND ITS APPLICATIONS

MATH48E FOURIER ANALYSIS AND ITS APPLICATIONS 7.. Cesàro summability. 7. Summability methods Arithmetic meas. The followig idea is due to the Italia geometer Eresto Cesàro (859-96). He shows that eve if

Sequences, 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

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;

On a Smarandache problem concerning the prime gaps

O a Smaradache problem cocerig the prime gaps Felice Russo Via A. Ifate 7 6705 Avezzao (Aq) Italy felice.russo@katamail.com Abstract I this paper, a problem posed i [] by Smaradache cocerig the prime gaps

Sequences. Notation. Convergence of a Sequence

Sequeces A sequece is essetially just a list. Defiitio (Sequece of Real Numbers). A sequece of real umbers is a fuctio Z (, ) R for some real umber. Do t let the descriptio of the domai cofuse you; it

JANE PROFESSOR WW Prob Lib1 Summer 2000

JANE PROFESSOR WW Prob Lib Summer 000 Sample WeBWorK problems. WeBWorK assigmet Series6CompTests due /6/06 at :00 AM..( pt) Test each of the followig series for covergece by either the Compariso Test or

Area As A Limit & Sigma Notation

Area As A Limit & Sigma Notatio SUGGESTED REFERENCE MATERIAL: As you work through the problems listed below, you should referece Chapter 5.4 of the recommeded textbook (or the equivalet chapter i your

September 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

IYGB. Special Extension Paper E. Time: 3 hours 30 minutes. Created by T. Madas. Created by T. Madas

YGB Special Extesio Paper E Time: 3 hours 30 miutes Cadidates may NOT use ay calculator. formatio for Cadidates This practice paper follows the Advaced Level Mathematics Core ad the Advaced Level Further

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,

5 Sequences and Series

Bria E. Veitch 5 Sequeces ad Series 5. Sequeces A sequece is a list of umbers i a defiite order. a is the first term a 2 is the secod term a is the -th term The sequece {a, a 2, a 3,..., a,..., } is a

UNIT #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

Area under a Curve-Using a Limit

Area uder a Curve-Usig a it Sice lettig be a very large umber will result i a huge amout of work, the process ca be simplified by usig sigma otatio ad summatio formulas to create a Riema Sum The ext example

arxiv: v2 [math.nt] 10 May 2014

FUNCTIONAL EQUATIONS RELATED TO THE DIRICHLET LAMBDA AND BETA FUNCTIONS JEONWON KIM arxiv:4045467v mathnt] 0 May 04 Abstract We give closed-form expressios for the Dirichlet beta fuctio at eve positive

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

Starting 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

AP Calculus Chapter 9: Infinite Series

AP Calculus Chapter 9: Ifiite Series 9. Sequeces a, a 2, a 3, a 4, a 5,... Sequece: A fuctio whose domai is the set of positive itegers = 2 3 4 a = a a 2 a 3 a 4 terms of the sequece Begi with the patter

Areas and Distances. We can easily find areas of certain geometric figures using well-known formulas:

Areas ad Distaces We ca easily fid areas of certai geometric figures usig well-kow formulas: However, it is t easy to fid the area of a regio with curved sides: METHOD: To evaluate the area of the regio

sin(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

a 3, a 4, ... are the terms of the sequence. The number a n is the nth term of the sequence, and the entire sequence is denoted by a n

60_090.qxd //0 : PM Page 59 59 CHAPTER 9 Ifiite Series Sectio 9. EXPLORATION Fidig Patters Describe a patter for each of the followig sequeces. The use your descriptio to write a formula for the th term

MH1101 AY1617 Sem 2. Question 1. NOT TESTED THIS TIME

MH AY67 Sem Questio. NOT TESTED THIS TIME ( marks Let R be the regio bouded by the curve y 4x x 3 ad the x axis i the first quadrat (see figure below. Usig the cylidrical shell method, fid the volume of

Sec 8.4. Alternating Series Test. A. Before Class Video Examples. Example 1: Determine whether the following series is convergent or divergent.

Sec 8.4 Alteratig Series Test A. Before Class Video Examples Example 1: Determie whether the followig series is coverget or diverget. a) ( 1)+1 =1 b) ( 1) 2 1 =1 Example 2: Determie whether the followig

Physics 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

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

In number theory we will generally be working with integers, though occasionally fractions and irrationals will come into play.

Number Theory Math 5840 otes. Sectio 1: Axioms. I umber theory we will geerally be workig with itegers, though occasioally fractios ad irratioals will come ito play. Notatio: Z deotes the set of all itegers

The Phi Power Series

The Phi Power Series I did this work i about 0 years while poderig the relatioship betwee the golde mea ad the Madelbrot set. I have fially decided to make it available from my blog at http://semresearch.wordpress.com/.

Revision 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

Carleton College, Winter 2017 Math 121, Practice Final Prof. Jones. Note: the exam will have a section of true-false questions, like the one below.

Carleto College, Witer 207 Math 2, Practice Fial Prof. Joes Note: the exam will have a sectio of true-false questios, like the oe below.. True or False. Briefly explai your aswer. A icorrectly justified

The multiplicative structure of finite field and a construction of LRC

IERG6120 Codig for Distributed Storage Systems Lecture 8-06/10/2016 The multiplicative structure of fiite field ad a costructio of LRC Lecturer: Keeth Shum Scribe: Zhouyi Hu Notatios: We use the otatio

Math 105: Review for Final Exam, Part II - SOLUTIONS

Math 5: Review for Fial Exam, Part II - SOLUTIONS. Cosider the fuctio f(x) = x 3 lx o the iterval [/e, e ]. (a) Fid the x- ad y-coordiates of ay ad all local extrema ad classify each as a local maximum

Math 2784 (or 2794W) University of Connecticut

ORDERS OF GROWTH PAT SMITH Math 2784 (or 2794W) Uiversity of Coecticut Date: Mar. 2, 22. ORDERS OF GROWTH. Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really

n=1 a n is the sequence (s n ) n 1 n=1 a n converges to s. We write a n = s, n=1 n=1 a n

Series. Defiitios ad first properties A series is a ifiite sum a + a + a +..., deoted i short by a. The sequece of partial sums of the series a is the sequece s ) defied by s = a k = a +... + a,. k= Defiitio

subcaptionfont+=small,labelformat=parens,labelsep=space,skip=6pt,list=0,hypcap=0 subcaption ALGEBRAIC COMBINATORICS LECTURE 8 TUESDAY, 2/16/2016

subcaptiofot+=small,labelformat=pares,labelsep=space,skip=6pt,list=0,hypcap=0 subcaptio ALGEBRAIC COMBINATORICS LECTURE 8 TUESDAY, /6/06. Self-cojugate Partitios Recall that, give a partitio λ, we may

Riemann Sums y = f (x)

Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, o-egative fuctio o the closed iterval [a, b] Fid

For use only in Badminton School November 2011 C2 Note. C2 Notes (Edexcel)

For use oly i Badmito School November 0 C Note C Notes (Edecel) Copyright www.pgmaths.co.uk - For AS, A otes ad IGCSE / GCSE worksheets For use oly i Badmito School November 0 C Note Copyright www.pgmaths.co.uk

Binomial distribution questions: formal word problems

Biomial distributio questios: formal word problems For the followig questios, write the iformatio give i a formal way before solvig the problem, somethig like: X = umber of... out of 2, so X B(2, 0.2).

CATHOLIC JUNIOR COLLEGE General Certificate of Education Advanced Level Higher 2 JC2 Preliminary Examination MATHEMATICS 9740/01

CATHOLIC JUNIOR COLLEGE Geeral Certificate of Educatio Advaced Level Higher JC Prelimiary Examiatio MATHEMATICS 9740/0 Paper 4 Aug 06 hours Additioal Materials: List of Formulae (MF5) Name: Class: READ

Polynomial Functions and Their Graphs

Polyomial Fuctios ad Their Graphs I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively

CSE 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

= = =

Sec 5.8 Sec 6. Mathematical Modelig (Arithmetic & Geometric Series) Name: Carl Friedrich Gauss is probably oe of the most oted complete mathematicias i history. As the story goes, he was potetially recogiized

CONTENTS. Course Goals. Course Materials Lecture Notes:

INTRODUCTION Ho Chi Mih City OF Uiversity ENVIRONMENTAL of Techology DESIGN Faculty Chapter of Civil 1: Orietatio. Egieerig Evaluatio Departmet of mathematical of Water Resources skill Egieerig & Maagemet

3. Z Transform. Recall that the Fourier transform (FT) of a DT signal xn [ ] is ( ) [ ] = In order for the FT to exist in the finite magnitude sense,

3. Z Trasform Referece: Etire Chapter 3 of text. Recall that the Fourier trasform (FT) of a DT sigal x [ ] is ω ( ) [ ] X e = j jω k = xe I order for the FT to exist i the fiite magitude sese, S = x [

Mathematical Foundations -1- Sets and Sequences. Sets and Sequences

Mathematical Foudatios -1- Sets ad Sequeces Sets ad Sequeces Methods of proof 2 Sets ad vectors 13 Plaes ad hyperplaes 18 Liearly idepedet vectors, vector spaces 2 Covex combiatios of vectors 21 eighborhoods,