8.1 Arithmetic Sequences
|
|
- Lenard Casey
- 7 years ago
- Views:
Transcription
1 MCR3U Uit 8: Sequeces & Series Page 1 of Arithmetic Sequeces Defiitio: A sequece is a comma separated list of ordered terms that follow a patter. Examples: 1, 2, 3, 4, 5 : a sequece of the first 5 itegers. 1, 4, 9, 16, : a sequece of the perfect squares startig at 1. The at the ed of the sequece is called the ellipsis ad idicates that the sequece cotiues idefiitely ,,,, : a sequece of fractios where deomiators are the itegers from Q1. Idetify the patter i each of the followig sequeces, ad the write the ext 2 terms: 1, 4, 7, 10, 13, 2, 4, 8, 16, ,,,, Formulas ad Notatio We use the followig letters ad otatio to represet iformatio about a sequece: : represets the term umber. The term umber must be a iteger startig at 1. t : represets the value of the th term. It is sometimes writte as t() which follows fuctio otatio. For example i the sequece 1, 4, 7, 10 the 7 is the 3 rd term, so we ca say that t 3 = 7. Notice that the subscript of 3 idicates that it is the third term. We ca use this otatio to defie formulas for sequeces based o the value of. Q2. Write the first three terms of each sequece, give the formula. a) t = b) t() = 2
2 MCR3U Uit 8: Sequeces & Series Page 2 of 2 Defiitio: A arithmetic sequece is ay sequece that shows a commo differece betwee terms. That is, the differece betwee ay two cosecutive terms i the sequece is a costat value. Also, if you kow a term i the sequece, you ca add this commo differece to it to determie the value of the ext term. Q3. Determie if each sequece is arithmetic, ad if it is, write the ext two terms. a) 4, 9, 14, 19, 24,, b) 10, 7, 4, 1, -2,, c) 2, 5, 8, 11, 15,, Formula for Arithmetic Sequeces Let a represet the first term of the sequece. (first letter of alphabet first term) Let d represet the commo differece for arithmetic sequeces. (d for differece) Write expressios for the first 4 terms of a arithmetic sequece, usig a ad d. t 1 = t 2 = t 3 = t 4 = Use ay patters you see above to write a formula for the th term t, usig a,, & d. t = Q4. For each of the followig sequeces, idetify the formula for the th term ad use your formula to determie the 100 th term i the sequece. a) 5, 7, 9, 11, b) 10, 7, 4, 1, c),,,,
3 MCR3U Uit 8: Sequeces & Series Page 3 of 3 Q5. Determie the umber of terms i the sequece 17, 14, 11, 8,, 136. Q6. I a arithmetic sequece, t 4 = 3 ad t 15 = 19. Determie the first three terms. Q7. Determie the value of x so that the three give terms form a arithmetic sequece (x 2), (x 4), (2x + 1).
4 MCR3U Uit 8: Sequeces & Series Page 4 of Geometric Sequeces Warmup Q1. Idetify the followig sequeces as arithmetic or o-arithmetic. If the sequece follows a patter, write the ext three terms. a) 1, 4, 9, 16,,, b) 2, 6, 10, 14,,, c) 3, 6, 12, 24,,, Q2. I a arithmetic sequece, t = 20 ad t 10 = 32. Determie: a) The first term ad commo differece. 6 b) The formula for the th term of the sequece. c) The value of t 100
5 MCR3U Uit 8: Sequeces & Series Page 5 of 5 Defiitio: A geometric sequece is ay sequece that shows a commo ratio betwee terms. That is, the ratio betwee ay two cosecutive terms i the sequece is a costat value. Also, if you kow a term i the sequece, you ca multiply the term by the commo ratio to determie the value of the ext term. Q1. Determie if each sequece is geometric, ad if it is, write the ext two terms. a) 2, 6, 18, 54,, b) 1024, 512, 256, 128,, c) 8, 16, 32, 64, 126,, Formula for Geometric Sequeces Let a represet the first term of the sequece. (first letter of alphabet first term) Let r represet the commo ratio for geometric sequeces. (r for ratio) Write expressios for the first 4 terms of a geometric sequece, usig a ad r. t 1 = t 2 = t 3 = t 4 = Use ay patters you see above to write a formula for the th term t, usig a,, & r. t = Q2. For each of the followig sequeces, idetify the formula for the th term ad use your formula to determie the 10 th term i the sequece. a) 2, 6, 18, b) 1024, 512, 256, c) 2, -2, 2, -2,
6 MCR3U Uit 8: Sequeces & Series Page 6 of 6 3 Q3. Determie the umber of terms i the sequece 48, 24, 12,,. 64 Q4. I a geometric sequece, t 5 = 405 ad t 9 = Determie the first three terms. Q5. Determie the value(s) of x so that the three give terms form a geometric sequece. (x 2), (2x 4), (x + 1).
7 MCR3U Uit 8: Sequeces & Series Page 7 of Recursio Formulas I previous lessos, we have determied sequeces usig a formula for the th term. A example is the formula t = 2 + 3, which determies the arithmetic sequece 5, 7, 9, 11, Aother example is t = 2 1, which determies the geometric sequece 1, 2, 4, 8, 16, Such formulas are kow as explicit formulas. They ca be used to calculate ay term i a sequece without kowig the previous term. For example, the teth term i the sequece determied by the formula t = is: t 10 = 2(10) + 3, which equals 23. It is sometimes more coveiet to calculate a term i a sequece from oe or more previous terms i the sequece. Formulas that ca be used to do this are called recursio formulas. A recursio formula cosists of at least two parts. The parts give the value(s) of the first term(s) i the sequece, ad a equatio that ca be used to calculate each of the other terms from the term(s) before it. A example is the formula: t 1 = 5 t = t The first part of the formula shows that the first term is 5. The secod part of the formula shows that each term after the first term is foud by addig 2 to the previous term. The sequece geerated will be 5, 7, 9, 11, 13, which happes to be a arithmetic sequece with explicit formula t = Q1. Write the first 5 terms for the sequece defied by: t t 1 = 1 = t 1 + Q2. Is the sequece i Q1 above arithmetic or geometric? Recursio Formulas for Arithmetic ad Geometric Sequeces Arithmetic Sequeces t = a t 1 = t 1 + d Geometric Sequeces t = a t 1 = t 1 r
8 MCR3U Uit 8: Sequeces & Series Page 8 of 8 Fiboacci Sequece Oe of the most famous recursive sequeces is the Fiboacci Sequece, which was discovered by a Italia mathematicia amed Leoardo Fiboacci (c ). It is defied as: t1 = 1 t2 = 1 t = t + 1 t 2 This says that: the first term is 1, the secod term is 1, ad the every term afterwards is the sum of the previous two terms. Q3. Write the first 12 terms of the Fiboacci sequece. Q4. Is the Fiboacci sequece arithmetic or geometric? Iterestig Iformatio about the Fiboacci Sequece While the Fiboacci sequece is ot geometric, there is a patter betwee cosecutive terms. As the sequece cotiues idefiitely, the ratio betwee cosecutive terms approaches a value of approximately (Try it with the umbers foud i Q3) This value is kow as the golde ratio ad has a exact value of This umber ca also be foud usig ifiite patters ad The golde ratio is the oly positive aswer to the questio What umber mius oe is equal to the reciprocal of the umber? The golde ratio has a lot of applicatios i mathematics, architecture, art, ad ature
9 MCR3U Uit 8: Sequeces & Series Page 9 of 9 Q5. Each stroke of a vacuum pump removes oe third of the air remaiig i the cotaier. What percet of the origial quatity of air remais i the cotaier after 10 strokes, to the earest percet? Summary of Sequeces Sequeces Terms t value of th term term umber i sequece a first term of sequece d commo differece r commo ratio Formulas Arithmetic Sequeces t = a + ( 1)d Geometric Sequeces t = a(r) -1
10 MCR3U Uit 8: Sequeces & Series Page 10 of Arithmetic Series Defiitio: A series is the sum of terms i a sequece. A arithmetic series is the sum of terms i a arithmetic sequece Notatio: S represets the sum of the first terms i a sequece. (S for Sum) Usig Gauss Method to fid the Sum ie. S = t 1 + t 2 + t t -1 + t Q1. Determie the sum of the first 100 atural umbers, usig the Gauss Method. Gauss Method rewrite the series with the terms reversed. add the two series together look for patters to simplify, the rearrage for S S 100 = Gauss Method usig Geeral Terms Q2. Use the Gauss Method, to determie the sum of the first terms of a arithmetic sequece. [Hit: keep t 1 ad t, but rewrite the other terms usig t 1, t, ad d.] S = t 1 + t 2 + t t -2 + t -1 + t
11 MCR3U Uit 8: Sequeces & Series Page 11 of 11 Formulas for Arithmetic Series S = ( t1 + t ) 2 or by replacig t 1 = a, ad t = a+(-1)d, we get = [ 2a + ( 1) d] S 2 The versio of the formula you use depeds o what iformatio you have. If you have t, use the first versio. If you have d, use the secod versio. Notice tha i either equatio, you must kow. Q3. Determie the sum of the first 150 terms of the arithmetic series Q4. Determie the sum of the first 50 terms of a arithmetic sequece defied by t = 4 1. Q5. Determie the sum of the arithmetic series:
12 MCR3U Uit 8: Sequeces & Series Page 12 of 12 Q6. The first 50 terms of a arithmetic series with commo differece of 6 are added together to get a sum of Determie the first ad last terms of this arithmetic sequece. Q7. The sum of the series = How may terms are i this series?
13 MCR3U Uit 8: Sequeces & Series Page 13 of Geometric Series Defiitio: A geometric series is the sum of terms i a geometric sequece. Determiig a Formula for Geometric Series Step 1. Write the sum of terms i a Geometric Series, usig a, r &. S = t 1 + t 2 + t t -2 + t -1 + t Step 2. Multiply the etire expressio by the commo ratio, r. Step 3. You should otice a iterestig patter betwee the equatios i steps 1 & 2. Subtract the equatio i step 1, from the equatio i step 2. Step 4. Simplify ad commo factor each side. Step 5. Rearrage the expressio to fid S.
14 MCR3U Uit 8: Sequeces & Series Page 14 of 14 Formulas for Geometric Series S = ( -1) a r ( r -1) This oly works if r 1. If r = 1, all the terms i the expressio are equal to the first term a, so the S = a 1 1 Q1. Determie the sum of the first 10 terms of the geometric series Q2. Determie the sum of the first 15 terms of a geometric sequece defied by t 3 = Q3. Determie the sum of the geometric series:
15 MCR3U Uit 8: Sequeces & Series Page 15 of 15 Q4. For a geometric sequece, S 1 = 2 ad S 2 = 8. Determie S 5. Q5. The sum of the series = How may terms are i this series? Q6. I a geometric sequece with commo ratio of 4, the sum of the first 7 terms is What is the first term of this sequece?
Infinite Sequences and Series
CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...
More informationBasic Elements of Arithmetic Sequences and Series
MA40S PRE-CALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic
More informationIn nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008
I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces
More information.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth
Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,
More informationSECTION 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 informationExample 2 Find the square root of 0. The only square root of 0 is 0 (since 0 is not positive or negative, so those choices don t exist here).
BEGINNING ALGEBRA Roots ad Radicals (revised summer, 00 Olso) Packet to Supplemet the Curret Textbook - Part Review of Square Roots & Irratioals (This portio ca be ay time before Part ad should mostly
More informationTHE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction
THE ARITHMETIC OF INTEGERS - multiplicatio, expoetiatio, divisio, additio, ad subtractio What to do ad what ot to do. THE INTEGERS Recall that a iteger is oe of the whole umbers, which may be either positive,
More informationTrigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is
0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values
More informationBINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients
652 (12-26) Chapter 12 Sequeces ad Series 12.5 BINOMIAL EXPANSIONS I this sectio Some Examples Otaiig the Coefficiets The Biomial Theorem I Chapter 5 you leared how to square a iomial. I this sectio you
More informationSEQUENCES AND SERIES CHAPTER
CHAPTER SEQUENCES AND SERIES Whe the Grat family purchased a computer for $,200 o a istallmet pla, they agreed to pay $00 each moth util the cost of the computer plus iterest had bee paid The iterest each
More informationSEQUENCES AND SERIES
Chapter 9 SEQUENCES AND SERIES Natural umbers are the product of huma spirit. DEDEKIND 9.1 Itroductio I mathematics, the word, sequece is used i much the same way as it is i ordiary Eglish. Whe we say
More informationSequences and Series
CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their
More informationNATIONAL SENIOR CERTIFICATE GRADE 12
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 04 MARKS: 50 TIME: 3 hours This questio paper cosists of 8 pages ad iformatio sheet. Please tur over Mathematics/P DBE/04 NSC Grade Eemplar INSTRUCTIONS
More informationSoving Recurrence Relations
Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree
More informationListing terms of a finite sequence List all of the terms of each finite sequence. a) a n n 2 for 1 n 5 1 b) a n for 1 n 4 n 2
74 (4 ) Chapter 4 Sequeces ad Series 4. SEQUENCES I this sectio Defiitio Fidig a Formula for the th Term The word sequece is a familiar word. We may speak of a sequece of evets or say that somethig is
More informationDepartment of Computer Science, University of Otago
Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly
More informationFIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix
FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if
More informationMATHEMATICS P1 COMMON TEST JUNE 2014 NATIONAL SENIOR CERTIFICATE GRADE 12
Mathematics/P1 1 Jue 014 Commo Test MATHEMATICS P1 COMMON TEST JUNE 014 NATIONAL SENIOR CERTIFICATE GRADE 1 Marks: 15 Time: ½ hours N.B: This questio paper cosists of 7 pages ad 1 iformatio sheet. Please
More informationIncremental calculation of weighted mean and variance
Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically
More informationCHAPTER 3 THE TIME VALUE OF MONEY
CHAPTER 3 THE TIME VALUE OF MONEY OVERVIEW A dollar i the had today is worth more tha a dollar to be received i the future because, if you had it ow, you could ivest that dollar ad ear iterest. Of all
More informationNATIONAL SENIOR CERTIFICATE GRADE 11
NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P NOVEMBER 007 MARKS: 50 TIME: 3 hours This questio paper cosists of 9 pages, diagram sheet ad a -page formula sheet. Please tur over Mathematics/P DoE/November
More informationI. Chi-squared Distributions
1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.
More informationCS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations
CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad
More informationYour organization has a Class B IP address of 166.144.0.0 Before you implement subnetting, the Network ID and Host ID are divided as follows:
Subettig Subettig is used to subdivide a sigle class of etwork i to multiple smaller etworks. Example: Your orgaizatio has a Class B IP address of 166.144.0.0 Before you implemet subettig, the Network
More information1 Correlation and Regression Analysis
1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio
More informationLesson 15 ANOVA (analysis of variance)
Outlie Variability -betwee group variability -withi group variability -total variability -F-ratio Computatio -sums of squares (betwee/withi/total -degrees of freedom (betwee/withi/total -mea square (betwee/withi
More informationFOUNDATIONS OF MATHEMATICS AND PRE-CALCULUS GRADE 10
FOUNDATIONS OF MATHEMATICS AND PRE-CALCULUS GRADE 10 [C] Commuicatio Measuremet A1. Solve problems that ivolve liear measuremet, usig: SI ad imperial uits of measure estimatio strategies measuremet strategies.
More informationProperties of MLE: consistency, asymptotic normality. Fisher information.
Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout
More informationHow To Solve The Homewor Problem Beautifully
Egieerig 33 eautiful Homewor et 3 of 7 Kuszmar roblem.5.5 large departmet store sells sport shirts i three sizes small, medium, ad large, three patters plaid, prit, ad stripe, ad two sleeve legths log
More informationSection 11.3: The Integral Test
Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult
More information5.3. Generalized Permutations and Combinations
53 GENERALIZED PERMUTATIONS AND COMBINATIONS 73 53 Geeralized Permutatios ad Combiatios 53 Permutatios with Repeated Elemets Assume that we have a alphabet with letters ad we wat to write all possible
More informationTHE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n
We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample
More informationDefinition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean
1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.
More informationMathematical goals. Starting points. Materials required. Time needed
Level A1 of challege: C A1 Mathematical goals Startig poits Materials required Time eeded Iterpretig algebraic expressios To help learers to: traslate betwee words, symbols, tables, ad area represetatios
More information3. Greatest Common Divisor - Least Common Multiple
3 Greatest Commo Divisor - Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd
More informationCHAPTER 3 DIGITAL CODING OF SIGNALS
CHAPTER 3 DIGITAL CODING OF SIGNALS Computers are ofte used to automate the recordig of measuremets. The trasducers ad sigal coditioig circuits produce a voltage sigal that is proportioal to a quatity
More informationMath C067 Sampling Distributions
Math C067 Samplig Distributios Sample Mea ad Sample Proportio Richard Beigel Some time betwee April 16, 2007 ad April 16, 2007 Examples of Samplig A pollster may try to estimate the proportio of voters
More informationMATH 083 Final Exam Review
MATH 08 Fial Eam Review Completig the problems i this review will greatly prepare you for the fial eam Calculator use is ot required, but you are permitted to use a calculator durig the fial eam period
More informationHere are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
This documet was writte ad copyrighted by Paul Dawkis. Use of this documet ad its olie versio is govered by the Terms ad Coditios of Use located at http://tutorial.math.lamar.edu/terms.asp. The olie versio
More informationElementary Theory of Russian Roulette
Elemetary Theory of Russia Roulette -iterestig patters of fractios- Satoshi Hashiba Daisuke Miematsu Ryohei Miyadera Itroductio. Today we are goig to study mathematical theory of Russia roulette. If some
More informationThe following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles
The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio
More information*The most important feature of MRP as compared with ordinary inventory control analysis is its time phasing feature.
Itegrated Productio ad Ivetory Cotrol System MRP ad MRP II Framework of Maufacturig System Ivetory cotrol, productio schedulig, capacity plaig ad fiacial ad busiess decisios i a productio system are iterrelated.
More informationMeasures of Spread and Boxplots Discrete Math, Section 9.4
Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,
More informationINFINITE SERIES KEITH CONRAD
INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal
More informationA probabilistic proof of a binomial identity
A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two
More informationUniversity of California, Los Angeles Department of Statistics. Distributions related to the normal distribution
Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chi-square (χ ) distributio.
More information5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?
5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso
More informationCS103X: Discrete Structures Homework 4 Solutions
CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible six-figure salaries i whole dollar amouts are there that cotai at least
More informationA Guide to the Pricing Conventions of SFE Interest Rate Products
A Guide to the Pricig Covetios of SFE Iterest Rate Products SFE 30 Day Iterbak Cash Rate Futures Physical 90 Day Bak Bills SFE 90 Day Bak Bill Futures SFE 90 Day Bak Bill Futures Tick Value Calculatios
More informationTheorems About Power Series
Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real o-egative umber R, called the radius
More informationhttp://www.webassign.net/v4cgijeff.downs@wnc/control.pl
Assigmet Previewer http://www.webassig.et/vcgijeff.dows@wc/cotrol.pl of // : PM Practice Eam () Questio Descriptio Eam over chapter.. Questio DetailsLarCalc... [] Fid the geeral solutio of the differetial
More informationLecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem
Lecture 4: Cauchy sequeces, Bolzao-Weierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits
More informationSAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx
SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval
More informationDiscrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13
EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may
More informationS. Tanny MAT 344 Spring 1999. be the minimum number of moves required.
S. Tay MAT 344 Sprig 999 Recurrece Relatios Tower of Haoi Let T be the miimum umber of moves required. T 0 = 0, T = 7 Iitial Coditios * T = T + $ T is a sequece (f. o itegers). Solve for T? * is a recurrece,
More information5 Boolean Decision Trees (February 11)
5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected
More informationWeek 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable
Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5
More informationTime Value of Money. First some technical stuff. HP10B II users
Time Value of Moey Basis for the course Power of compoud iterest $3,600 each year ito a 401(k) pla yields $2,390,000 i 40 years First some techical stuff You will use your fiacial calculator i every sigle
More informationChapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions
Chapter 5 Uit Aual Amout ad Gradiet Fuctios IET 350 Egieerig Ecoomics Learig Objectives Chapter 5 Upo completio of this chapter you should uderstad: Calculatig future values from aual amouts. Calculatig
More informationMARTINGALES AND A BASIC APPLICATION
MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measure-theoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this
More informationSolving equations. Pre-test. Warm-up
Solvig equatios 8 Pre-test Warm-up We ca thik of a algebraic equatio as beig like a set of scales. The two sides of the equatio are equal, so the scales are balaced. If we add somethig to oe side of the
More information7.1 Finding Rational Solutions of Polynomial Equations
4 Locker LESSON 7. Fidig Ratioal Solutios of Polyomial Equatios Name Class Date 7. Fidig Ratioal Solutios of Polyomial Equatios Essetial Questio: How do you fid the ratioal roots of a polyomial equatio?
More informationConfidence Intervals for One Mean
Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a
More informationLecture 4: Cheeger s Inequality
Spectral Graph Theory ad Applicatios WS 0/0 Lecture 4: Cheeger s Iequality Lecturer: Thomas Sauerwald & He Su Statemet of Cheeger s Iequality I this lecture we assume for simplicity that G is a d-regular
More informationPart - I. Mathematics
Part - I Mathematics CHAPTER Set Theory. Objectives. Itroductio. Set Cocept.. Sets ad Elemets. Subset.. Proper ad Improper Subsets.. Equality of Sets.. Trasitivity of Set Iclusio.4 Uiversal Set.5 Complemet
More informationSimple Annuities Present Value.
Simple Auities Preset Value. OBJECTIVES (i) To uderstad the uderlyig priciple of a preset value auity. (ii) To use a CASIO CFX-9850GB PLUS to efficietly compute values associated with preset value auities.
More information4.3. The Integral and Comparison Tests
4.3. THE INTEGRAL AND COMPARISON TESTS 9 4.3. The Itegral ad Compariso Tests 4.3.. The Itegral Test. Suppose f is a cotiuous, positive, decreasig fuctio o [, ), ad let a = f(). The the covergece or divergece
More information1 Computing the Standard Deviation of Sample Means
Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.
More informationChapter 5 O A Cojecture Of Erdíos Proceedigs NCUR VIII è1994è, Vol II, pp 794í798 Jeærey F Gold Departmet of Mathematics, Departmet of Physics Uiversity of Utah Do H Tucker Departmet of Mathematics Uiversity
More informationAsymptotic Growth of Functions
CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll
More informationLesson 17 Pearson s Correlation Coefficient
Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) -types of data -scatter plots -measure of directio -measure of stregth Computatio -covariatio of X ad Y -uique variatio i X ad Y -measurig
More informationConvexity, Inequalities, and Norms
Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for
More information1. MATHEMATICAL INDUCTION
1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1
More informationCS100: Introduction to Computer Science
I-class Exercise: CS100: Itroductio to Computer Sciece What is a flip-flop? What are the properties of flip-flops? Draw a simple flip-flop circuit? Lecture 3: Data Storage -- Mass storage & represetig
More informationEGYPTIAN FRACTION EXPANSIONS FOR RATIONAL NUMBERS BETWEEN 0 AND 1 OBTAINED WITH ENGEL SERIES
EGYPTIAN FRACTION EXPANSIONS FOR RATIONAL NUMBERS BETWEEN 0 AND OBTAINED WITH ENGEL SERIES ELVIA NIDIA GONZÁLEZ AND JULIA BERGNER, PHD DEPARTMENT OF MATHEMATICS Abstract. The aciet Egyptias epressed ratioal
More informationDetermining the sample size
Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors
More informationNormal Distribution.
Normal Distributio www.icrf.l Normal distributio I probability theory, the ormal or Gaussia distributio, is a cotiuous probability distributio that is ofte used as a first approimatio to describe realvalued
More informationPresent Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value
Cocept 9: Preset Value Is the value of a dollar received today the same as received a year from today? A dollar today is worth more tha a dollar tomorrow because of iflatio, opportuity cost, ad risk Brigig
More information3. If x and y are real numbers, what is the simplified radical form
lgebra II Practice Test Objective:.a. Which is equivalet to 98 94 4 49?. Which epressio is aother way to write 5 4? 5 5 4 4 4 5 4 5. If ad y are real umbers, what is the simplified radical form of 5 y
More informationhp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation
HP 1C Statistics - average ad stadard deviatio Average ad stadard deviatio cocepts HP1C average ad stadard deviatio Practice calculatig averages ad stadard deviatios with oe or two variables HP 1C Statistics
More information2-3 The Remainder and Factor Theorems
- The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x
More informationChapter 7: Confidence Interval and Sample Size
Chapter 7: Cofidece Iterval ad Sample Size Learig Objectives Upo successful completio of Chapter 7, you will be able to: Fid the cofidece iterval for the mea, proportio, ad variace. Determie the miimum
More informationSolving Logarithms and Exponential Equations
Solvig Logarithms ad Epoetial Equatios Logarithmic Equatios There are two major ideas required whe solvig Logarithmic Equatios. The first is the Defiitio of a Logarithm. You may recall from a earlier topic:
More informationRepeating Decimals are decimal numbers that have number(s) after the decimal point that repeat in a pattern.
5.5 Fractios ad Decimals Steps for Chagig a Fractio to a Decimal. Simplify the fractio, if possible. 2. Divide the umerator by the deomiator. d d Repeatig Decimals Repeatig Decimals are decimal umbers
More informationDescriptive Statistics
Descriptive Statistics We leared to describe data sets graphically. We ca also describe a data set umerically. Measures of Locatio Defiitio The sample mea is the arithmetic average of values. We deote
More informationLecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)
18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the Bru-Mikowski iequality for boxes. Today we ll go over the
More informationChapter 6: Variance, the law of large numbers and the Monte-Carlo method
Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value
More informationLearning objectives. Duc K. Nguyen - Corporate Finance 21/10/2014
1 Lecture 3 Time Value of Moey ad Project Valuatio The timelie Three rules of time travels NPV of a stream of cash flows Perpetuities, auities ad other special cases Learig objectives 2 Uderstad the time-value
More informationa 4 = 4 2 4 = 12. 2. Which of the following sequences converge to zero? n 2 (a) n 2 (b) 2 n x 2 x 2 + 1 = lim x n 2 + 1 = lim x
0 INFINITE SERIES 0. Sequeces Preiary Questios. What is a 4 for the sequece a? solutio Substitutig 4 i the expressio for a gives a 4 4 4.. Which of the followig sequeces coverge to zero? a b + solutio
More informationChapter 5: Inner Product Spaces
Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples
More informationChapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:
Chapter 7 - Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries
More informationI. Why is there a time value to money (TVM)?
Itroductio to the Time Value of Moey Lecture Outlie I. Why is there the cocept of time value? II. Sigle cash flows over multiple periods III. Groups of cash flows IV. Warigs o doig time value calculatios
More informationAP Calculus BC 2003 Scoring Guidelines Form B
AP Calculus BC Scorig Guidelies Form B The materials icluded i these files are iteded for use by AP teachers for course ad exam preparatio; permissio for ay other use must be sought from the Advaced Placemet
More informationGCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number.
GCSE STATISTICS You should kow: 1) How to draw a frequecy diagram: e.g. NUMBER TALLY FREQUENCY 1 3 5 ) How to draw a bar chart, a pictogram, ad a pie chart. 3) How to use averages: a) Mea - add up all
More informationGCE Further Mathematics (6360) Further Pure Unit 2 (MFP2) Textbook. Version: 1.4
GCE Further Mathematics (660) Further Pure Uit (MFP) Tetbook Versio: 4 MFP Tetbook A-level Further Mathematics 660 Further Pure : Cotets Chapter : Comple umbers 4 Itroductio 5 The geeral comple umber 5
More informationRemarques sur un beau rapport entre les series des puissances tant directes que reciproques
Aug 006 Traslatio with otes of Euler s paper Remarques sur u beau rapport etre les series des puissaces tat directes que reciproques Origially published i Memoires de l'academie des scieces de Berli 7
More informationAP Calculus AB 2006 Scoring Guidelines Form B
AP Calculus AB 6 Scorig Guidelies Form B The College Board: Coectig Studets to College Success The College Board is a ot-for-profit membership associatio whose missio is to coect studets to college success
More informationLaws of Exponents Learning Strategies
Laws of Epoets Learig Strategies What should studets be able to do withi this iteractive? Studets should be able to uderstad ad use of the laws of epoets. Studets should be able to simplify epressios that
More informationEngineering Data Management
BaaERP 5.0c Maufacturig Egieerig Data Maagemet Module Procedure UP128A US Documetiformatio Documet Documet code : UP128A US Documet group : User Documetatio Documet title : Egieerig Data Maagemet Applicatio/Package
More information