SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES"

Transcription

1 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, material ot icluded i the text. Cocept Developmet 1. I Sectio 1.5, we itroduce the cocept of summatio. I would like to begi with a story about Carl Freidrich Gauss ( ), sometimes rated as oe of the greatest mathematicias who ever lived. The story goes somethig like this: As a youth, Gauss of course atteded class with oly boys uder the tutelage of a headmaster. Oe day the boys were beig particularly uruly, so the master assiged them a task to keep them occupied: Add the umbers (that is ). The boys bet to their task, all but Carl Freidrich Gauss, who put his had i the air ad said Sir! The aswer is ad he was right! How did he do it? Oe way is to simply strig the umbers out as see below: If you look at the first ad last you see a sum of 101. If you look at the d ad the ext to last you see + 99 = 101. If you look at the ext pairig, , you see a sum of 101! There are 50 pairs i this overall sum ad each has a total of 101. The result: = 5050! Why is this process valid? If you start at the lower ed ad move i oe etry i the sum, that is, move from 1 to, you add 1. If you start at the upper ed ad move back oe i the sum, that is, move from 100 to 99, you lose 1. The loss o the upper ed compesates for the icrease o the lower ed. This process repeats itself as you move from oe pairig to aother throughout the overall sum. Result: 50 pairs of 101 ad total of This process works best whe you have a eve umber of terms i your sum. A alterative approach is to write the sum dow twice, oce frot to back ad the secod time back to frot as see below: This time we pair the umbers vertically, agai obtaiig sums of 101. How may such sums do we have? (Aswer: 100) Thus, i all the idividual sums we have a total of 100(101) = 10,100. However, we have couted each umber twice, so, to compesate, we divide by. Result? 5050! This process ca be geeralized with a sum of the form ( 1) + ( ) +

2 where you have terms. Whe you write the sum dow twice, oce frot to back ad the secod time back to frot, the add vertically, you have pairs of ( + 1) as see below ( ) + ( 1) + + ( + 1) + ( + ) Total? ( + 1). However we couted each term i our iitial sum twice, so, to compesate, we divide by, leadig to the formula: ( 1) + ( ) + = ( + 1) Thus, ay time we see a sum of this form, we may use the formula, the first of those preseted o page 8.. We ofte see sums represeted with summatio otatio. For example the sum of Gauss story may be represeted as: where the symbol the bouds of summatio = i stads for summatio, i is kow as a idex, ad the umbers 1 ad 100 are 10 You try oe! What sum is represeted by j? (Aswer: ) i = See Examples 1 ad, pages ad 7 for further work with these cocepts. Note also properties that apply illustrated o page 7 umbered to the right of the page, 7 ad Additioal formulas, preseted without proof at the top of page 8, are as follows: 100 i = = ( 1) ( + 1) + ad i 3 = = ( + 1) 4 Please see additioal work with these formulas ad the summatio properties i Example 3, pages 8 ad Next, we iclude some additioal material ot preseted i the text. What if we looked at the terms of a summatio, but did ot add? Such a listig of terms is kow as a sequece or a progressio. Examples below: 1,, 3, 4, a ifiite sequece sice the dots idicate it cotiues forever or

3 1,, 3, 4, , 100 a sequece represetig the terms of the sum we discussed above Obviously, terms of a sequece may be added. We may use summatio otatio or formulas ad properties discussed above whe appropriate. Such a summatio of the terms of a sequece is kow as a series. 5. We are ofte give a geeral term or th term for a sequece. For example, if I told you that you to list etries i a sequece with geeral term, you would respod with the listig, 4,, ad if I asked for etries i a sequece with geeral term, you would respod with 1, 4, 9, 1, 5,..... What about the sum of the first 10 etries of such a sequece? Use the formula preseted i step 3 above: ( 0 1) 10 (10 + 1) + = You would the evaluate usig the arithmetic tools of cacellatio ad calculatio to obtai There are two special types of sequeces, arithmetic ad geometric. (Note the ew use of familiar termiology) I a arithmetic sequece, we start with a iitial etry, the add a commo differece repeatedly obtaiig additioal etries i the sequece. If our iitial etry is ad our commo differece is 3, we would obtai the sequece, 5, 8, 11, 14, If our iitial etry was 3 ad our commo differece is, we would obtai the sequece 3, 5, 7, 9, 11, I a geometric sequece, we start with a iitial etry, the multiply by a commo ratio repeatedly obtaiig additioal etries i the sequece. If our iitial etry is ad our commo ratio is 3, we would obtai the sequece,, 18, 54, 1, If our iitial etry was 3 ad our commo ratio is, we would obtai the sequece 3,, 1, 4, 48, We are ofte give a arithmetic or geometric sequece ad asked to fid the geeral or th term. How do we go about this task?? I a very orgaized way! Let s start with the sequece 1, 3, 5, 7, 9, We have show the first terms i a ifiite sequece. The first term is 1, the d is 3, the 3 rd is 5 ad so o. Thus, you see that we may umber our terms. Let s put each term vertically ito a table, umber it ad aalyze it. Number of Term Aalysis Simplificatio Etry () () () () () ()

4 How ca we develop the geeral term from this iformatio? We ote, i the right had colum, that the iitial etry (1) ad the commo differece () are costat. The oly piece that chages is the coefficiet (multiplier) of. Ca we tie it to aythig? Yes! I each case the coefficiet is oe less tha the umber of the term! It would appear that our terms are of the form 1 + ( 1). Simplified, our geeral term is 1 + or 1. Lets check this! Our th term above is ad () 1 = 11! It appears to work! Ca we carry this forward? What would the 100 th term i our sequece be? (100) 1 = 199. What would the 1000 th etry be? (1000) 1 = 1999! 7. What about the geeral term of a geometric sequece? Let s start with the sequece 3,, 1, 4, 48, We have show the first terms i a ifiite sequece. The first term is 3, the d is, the 3 rd is 1 ad so o. Thus, you see that we may umber our terms. Let s put each term vertically ito a table, umber it ad aalyze it. Number of Term Aalysis Simplificatio Etry () ( 0 ) 3 3 ( 1 ) ( ) ( 3 ) ( 4 ) How ca we develop the geeral term from this iformatio? We ote, i the right had colum, that the iitial etry (3) ad the commo ratio () are costat. The oly piece that chages is the expoet applied to. Ca we tie it to aythig? Yes! I each case the expoet is oe less tha the umber of the term! It would appear that our terms are of the form 3( 1 ). Lets check this! Our th term above is ad 3( 5 ) = 3 3 = 9! It appears to work! Ca we carry this forward? What would the 100 th term i our sequece be?. 3( 99 ) =?!? (Let s leave it i expoetial form!) What would the 1000 th etry be? 3( 999 ) Do you eed additioal help with these cocepts? Help optios: 1. Try the practice problems below. Solutios are preseted followig the assigmet. A. Use a formula preseted i Sectio 1.5 to fid B. Evaluate j j = 1 C. Fid the 4 th term i the sequece with geeral term 3( 5 ( 1) ) D. Fid the geeral term of the arithmetic sequece 4, 7, 10, 13,

5 Summary I this sectio ad the additioal material, we leared work with summatio cocepts, formulae ad otatio ad with sequeces, usig a geeral term or format to develop a sequece ad fidig the geeral term of a give arithmetic or geometric sequece. Assigmet: Complete the followig Sectio 1.5 problems (page 9): 1 15 (odd problems oly), 19, plus additioal problems below. 1. If a sequece is represeted as a i =, 3, 4, 5,, fid a 3.. Write the first 5 terms of a arithmetic sequece with first term of 3 ad commo differece. 3. Write the first 5 terms of a geometric sequece with first term 3 ad commo ratio. 4. What are the ext 3 etries i the ifiite sequece 5, 10, 17,, 37,....? 5. a.) Fid the geeral term of the ifiite sequece 3, 8, 13, 18, 3, b.) Fid the 100 th term of this sequece.. a.) Fid the geeral term of the ifiite sequece 3,, 1, 4, 48, b.) Fid the 100 th term of this sequece. (You may leave your aswer i expoetial form.) Solutio to Practice Problem A: The formula of choice is 45 45(4) i = = ( + 1) i =. I this applicatio, it becomes Solutio to Practice Problem B: The formula of choice is i ( + 1)( = + 1). I this applicatio, we see 5 5( )( 51) 5( )( 17) i = = = (Note the use of the basic math techique cacellatio. ) 1 i= 1

6 Solutio to Practice Problem C: The 4 th term i the sequece with geeral term 3 5 ( 1) is 3 5 (4 1) = = 3 15 = 375 Solutio to Practice Problem D: The geeral term of the arithmetic sequece 4, 7, 10, 13, may be foud as show below. First umber the terms of the sequece with 4 as the 1 st, 7 as the d ad so o, where represets the umber of the geeral term. Term Aalysis of Term Simplificatio (3) (3) (3) (3) Thus we see that each term i the sequece is 4 + ( 1)(3) where is the umber of the term. Thus the geeral term is 4 + ( 1)(3) = = Check your work. Is 7 = 3() + 1? Yes! Is 10 = 3(3) + 1? Yes! ad so o.

8.1 Arithmetic Sequences

8.1 Arithmetic Sequences MCR3U Uit 8: Sequeces & Series Page 1 of 1 8.1 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

More information

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

In 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

Module 4: Mathematical Induction

Module 4: Mathematical Induction Module 4: Mathematical Iductio Theme 1: Priciple of Mathematical Iductio Mathematical iductio is used to prove statemets about atural umbers. As studets may remember, we ca write such a statemet as a predicate

More information

Geometric Sequences and Series. Geometric Sequences. Definition of Geometric Sequence. such that. a2 4

Geometric Sequences and Series. Geometric Sequences. Definition of Geometric Sequence. such that. a2 4 3330_0903qxd /5/05 :3 AM Page 663 Sectio 93 93 Geometric Sequeces ad Series 663 Geometric Sequeces ad Series What you should lear Recogize, write, ad fid the th terms of geometric sequeces Fid th partial

More information

Basic Elements of Arithmetic Sequences and Series

Basic 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 information

Section 9.2 Series and Convergence

Section 9.2 Series and Convergence Sectio 9. Series ad Covergece Goals of Chapter 9 Approximate Pi Prove ifiite series are aother importat applicatio of limits, derivatives, approximatio, slope, ad cocavity of fuctios. Fid challegig atiderivatives

More information

1 The Binomial Theorem: Another Approach

1 The Binomial Theorem: Another Approach The Biomial Theorem: Aother Approach Pascal s Triagle I class (ad i our text we saw that, for iteger, the biomial theorem ca be stated (a + b = c a + c a b + c a b + + c ab + c b, where the coefficiets

More information

Section IV.5: Recurrence Relations from Algorithms

Section IV.5: Recurrence Relations from Algorithms Sectio IV.5: Recurrece Relatios from Algorithms Give a recursive algorithm with iput size, we wish to fid a Θ (best big O) estimate for its ru time T() either by obtaiig a explicit formula for T() or by

More information

ARITHMETIC AND GEOMETRIC PROGRESSIONS

ARITHMETIC AND GEOMETRIC PROGRESSIONS Arithmetic Ad Geometric Progressios Sequeces Ad ARITHMETIC AND GEOMETRIC PROGRESSIONS Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives

More information

CHAPTER 3 THE TIME VALUE OF MONEY

CHAPTER 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 information

Radicals and Fractional Exponents

Radicals and Fractional Exponents Radicals ad Roots Radicals ad Fractioal Expoets I math, may problems will ivolve what is called the radical symbol, X is proouced the th root of X, where is or greater, ad X is a positive umber. What it

More information

THE ARITHMETIC OF INTEGERS. - multiplication, exponentiation, division, addition, and subtraction

THE 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 information

Soving Recurrence Relations

Soving 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 information

when n = 1, 2, 3, 4, 5, 6, This list represents the amount of dollars you have after n days. Note: The use of is read as and so on.

when n = 1, 2, 3, 4, 5, 6, This list represents the amount of dollars you have after n days. Note: The use of is read as and so on. Geometric eries Before we defie what is meat by a series, we eed to itroduce a related topic, that of sequeces. Formally, a sequece is a fuctio that computes a ordered list. uppose that o day 1, you have

More information

The second difference is the sequence of differences of the first difference sequence, 2

The second difference is the sequence of differences of the first difference sequence, 2 Differece Equatios I differetial equatios, you look for a fuctio that satisfies ad equatio ivolvig derivatives. I differece equatios, istead of a fuctio of a cotiuous variable (such as time), we look for

More information

BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients

BINOMIAL 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 information

The Euler Totient, the Möbius and the Divisor Functions

The Euler Totient, the Möbius and the Divisor Functions The Euler Totiet, the Möbius ad the Divisor Fuctios Rosica Dieva July 29, 2005 Mout Holyoke College South Hadley, MA 01075 1 Ackowledgemets This work was supported by the Mout Holyoke College fellowship

More information

.04. This means $1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth

.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 information

Example 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).

Example 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 information

Lesson 12. Sequences and Series

Lesson 12. Sequences and Series Retur to List of Lessos Lesso. Sequeces ad Series A ifiite sequece { a, a, a,... a,...} ca be thought of as a list of umbers writte i defiite order ad certai patter. It is usually deoted by { a } =, or

More information

4.1 Sigma Notation and Riemann Sums

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

More information

5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?

5.4 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 information

ORDERS OF GROWTH KEITH CONRAD

ORDERS OF GROWTH KEITH CONRAD ORDERS OF GROWTH KEITH CONRAD Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really wat to uderstad their behavior It also helps you better grasp topics i calculus

More information

A Gentle Introduction to Algorithms: Part II

A Gentle Introduction to Algorithms: Part II A Getle Itroductio to Algorithms: Part II Cotets of Part I:. Merge: (to merge two sorted lists ito a sigle sorted list.) 2. Bubble Sort 3. Merge Sort: 4. The Big-O, Big-Θ, Big-Ω otatios: asymptotic bouds

More information

Here 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.

Here 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 information

Definition. 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

Definition. 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 information

Infinite Sequences and Series

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 information

Lesson 17 Pearson s Correlation Coefficient

Lesson 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 information

hp calculators HP 30S Base Conversions Numbers in Different Bases Practice Working with Numbers in Different Bases

hp calculators HP 30S Base Conversions Numbers in Different Bases Practice Working with Numbers in Different Bases Numbers i Differet Bases Practice Workig with Numbers i Differet Bases Numbers i differet bases Our umber system (called Hidu-Arabic) is a decimal system (it s also sometimes referred to as deary system)

More information

Arithmetic Sequences and Partial Sums. Arithmetic Sequences. Definition of Arithmetic Sequence. Example 1. 7, 11, 15, 19,..., 4n 3,...

Arithmetic Sequences and Partial Sums. Arithmetic Sequences. Definition of Arithmetic Sequence. Example 1. 7, 11, 15, 19,..., 4n 3,... 3330_090.qxd 1/5/05 11:9 AM Page 653 Sectio 9. Arithmetic Sequeces ad Partial Sums 653 9. Arithmetic Sequeces ad Partial Sums What you should lear Recogize,write, ad fid the th terms of arithmetic sequeces.

More information

Recursion and Recurrences

Recursion and Recurrences Chapter 5 Recursio ad Recurreces 5.1 Growth Rates of Solutios to Recurreces Divide ad Coquer Algorithms Oe of the most basic ad powerful algorithmic techiques is divide ad coquer. Cosider, for example,

More information

Review for College Algebra Final Exam

Review for College Algebra Final Exam Review for College Algebra Fial Exam (Please remember that half of the fial exam will cover chapters 1-4. This review sheet covers oly the ew material, from chapters 5 ad 7.) 5.1 Systems of equatios i

More information

Searching Algorithm Efficiencies

Searching Algorithm Efficiencies Efficiecy of Liear Search Searchig Algorithm Efficiecies Havig implemeted the liear search algorithm, how would you measure its efficiecy? A useful measure (or metric) should be geeral, applicable to ay

More information

The geometric series and the ratio test

The geometric series and the ratio test The geometric series ad the ratio test Today we are goig to develop aother test for covergece based o the iterplay betwee the it compariso test we developed last time ad the geometric series. A ote about

More information

Incremental calculation of weighted mean and variance

Incremental 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 information

2.3. GEOMETRIC SERIES

2.3. GEOMETRIC SERIES 6 CHAPTER INFINITE SERIES GEOMETRIC SERIES Oe of the most importat types of ifiite series are geometric series A geometric series is simply the sum of a geometric sequece, Fortuately, geometric series

More information

Fourier Series and the Wave Equation Part 2

Fourier Series and the Wave Equation Part 2 Fourier Series ad the Wave Equatio Part There are two big ideas i our work this week. The first is the use of liearity to break complicated problems ito simple pieces. The secod is the use of the symmetries

More information

Page 2 of 14 = T(-2) + 2 = [ T(-3)+1 ] + 2 Substitute T(-3)+1 for T(-2) = T(-3) + 3 = [ T(-4)+1 ] + 3 Substitute T(-4)+1 for T(-3) = T(-4) + 4 After i

Page 2 of 14 = T(-2) + 2 = [ T(-3)+1 ] + 2 Substitute T(-3)+1 for T(-2) = T(-3) + 3 = [ T(-4)+1 ] + 3 Substitute T(-4)+1 for T(-3) = T(-4) + 4 After i Page 1 of 14 Search C455 Chapter 4 - Recursio Tree Documet last modified: 02/09/2012 18:42:34 Uses: Use recursio tree to determie a good asymptotic boud o the recurrece T() = Sum the costs withi each level

More information

Chapter 5 Unit 1. IET 350 Engineering Economics. Learning Objectives Chapter 5. Learning Objectives Unit 1. Annual Amount and Gradient Functions

Chapter 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 information

Listing 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

Listing 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 information

Confidence Intervals for One Mean with Tolerance Probability

Confidence Intervals for One Mean with Tolerance Probability Chapter 421 Cofidece Itervals for Oe Mea with Tolerace Probability Itroductio This procedure calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) with

More information

Riemann Sums y = f (x)

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

More information

hp calculators HP 12C Statistics - average and standard deviation Average and standard deviation concepts HP12C average and standard deviation

hp 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 information

I. Chi-squared Distributions

I. 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 information

Sequences and Series

Sequences 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 information

Math 475, Problem Set #6: Solutions

Math 475, Problem Set #6: Solutions Math 475, Problem Set #6: Solutios A (a) For each poit (a, b) with a, b o-egative itegers satisfyig ab 8, cout the paths from (0,0) to (a, b) where the legal steps from (i, j) are to (i 2, j), (i, j 2),

More information

TILE PATTERNS & GRAPHING

TILE PATTERNS & GRAPHING TILE PATTERNS & GRAPHING LESSON 1 THE BIG IDEA Tile patters provide a meaigful cotext i which to geerate equivalet algebraic expressios ad develop uderstadig of the cocept of a variable. Such patters are

More information

Divide and Conquer, Solving Recurrences, Integer Multiplication Scribe: Juliana Cook (2015), V. Williams Date: April 6, 2016

Divide and Conquer, Solving Recurrences, Integer Multiplication Scribe: Juliana Cook (2015), V. Williams Date: April 6, 2016 CS 6, Lecture 3 Divide ad Coquer, Solvig Recurreces, Iteger Multiplicatio Scribe: Juliaa Cook (05, V Williams Date: April 6, 06 Itroductio Today we will cotiue to talk about divide ad coquer, ad go ito

More information

Sequences II. Chapter 3. 3.1 Convergent Sequences

Sequences II. Chapter 3. 3.1 Convergent Sequences Chapter 3 Sequeces II 3. Coverget Sequeces Plot a graph of the sequece a ) = 2, 3 2, 4 3, 5 + 4,...,,... To what limit do you thik this sequece teds? What ca you say about the sequece a )? For ǫ = 0.,

More information

1.3 Binomial Coefficients

1.3 Binomial Coefficients 18 CHAPTER 1. COUNTING 1. Biomial Coefficiets I this sectio, we will explore various properties of biomial coefficiets. Pascal s Triagle Table 1 cotais the values of the biomial coefficiets ( ) for 0to

More information

1 n. n > dt. t < n 1 + n=1

1 n. n > dt. t < n 1 + n=1 Math 05 otes C. Pomerace The harmoic sum The harmoic sum is the sum of recirocals of the ositive itegers. We kow from calculus that it diverges, this is usually doe by the itegral test. There s a more

More information

CS103X: Discrete Structures Homework 4 Solutions

CS103X: 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 information

Learning outcomes. Algorithms and Data Structures. Time Complexity Analysis. Time Complexity Analysis How fast is the algorithm? Prof. Dr.

Learning outcomes. Algorithms and Data Structures. Time Complexity Analysis. Time Complexity Analysis How fast is the algorithm? Prof. Dr. Algorithms ad Data Structures Algorithm efficiecy Learig outcomes Able to carry out simple asymptotic aalysisof algorithms Prof. Dr. Qi Xi 2 Time Complexity Aalysis How fast is the algorithm? Code the

More information

Chapter One BASIC MATHEMATICAL TOOLS

Chapter One BASIC MATHEMATICAL TOOLS Chapter Oe BAIC MATHEMATICAL TOOL As the reader will see, the study of the time value of moey ivolves substatial use of variables ad umbers that are raised to a power. The power to which a variable is

More information

Chapter Gaussian Elimination

Chapter Gaussian Elimination Chapter 04.06 Gaussia Elimiatio After readig this chapter, you should be able to:. solve a set of simultaeous liear equatios usig Naïve Gauss elimiatio,. lear the pitfalls of the Naïve Gauss elimiatio

More information

Lesson 15 ANOVA (analysis of variance)

Lesson 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 information

Mocks.ie Maths LC HL Further Calculus mocks.ie Page 1

Mocks.ie Maths LC HL Further Calculus mocks.ie Page 1 Maths Leavig Cert Higher Level Further Calculus Questio Paper By Cillia Fahy ad Darro Higgis Mocks.ie Maths LC HL Further Calculus mocks.ie Page Further Calculus ad Series, Paper II Q8 Table of Cotets:.

More information

Algebra Work Sheets. Contents

Algebra Work Sheets. Contents The work sheets are grouped accordig to math skill. Each skill is the arraged i a sequece of work sheets that build from simple to complex. Choose the work sheets that best fit the studet s eed ad will

More information

Divide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015

Divide and Conquer. Maximum/minimum. Integer Multiplication. CS125 Lecture 4 Fall 2015 CS125 Lecture 4 Fall 2015 Divide ad Coquer We have see oe geeral paradigm for fidig algorithms: the greedy approach. We ow cosider aother geeral paradigm, kow as divide ad coquer. We have already see a

More information

THE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n

THE 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 information

Confidence Intervals for One Mean

Confidence 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 information

I. Why is there a time value to money (TVM)?

I. 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 information

A Resource for Free-standing Mathematics Qualifications Working with %

A Resource for Free-standing Mathematics Qualifications Working with % Ca you aswer these questios? A savigs accout gives % iterest per aum.. If 000 is ivested i this accout, how much will be i the accout at the ed of years? A ew car costs 16 000 ad its value falls by 1%

More information

Engineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51

Engineering 323 Beautiful Homework Set 3 1 of 7 Kuszmar Problem 2.51 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 information

Asymptotic Growth of Functions

Asymptotic 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 information

Section 11.3: The Integral Test

Section 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 information

Math Background: Review & Beyond. Design & Analysis of Algorithms COMP 482 / ELEC 420. Solving for Closed Forms. Obtaining Recurrences

Math Background: Review & Beyond. Design & Analysis of Algorithms COMP 482 / ELEC 420. Solving for Closed Forms. Obtaining Recurrences Math Backgroud: Review & Beyod. Asymptotic otatio Desig & Aalysis of Algorithms COMP 48 / ELEC 40 Joh Greier. Math used i asymptotics 3. Recurreces 4. Probabilistic aalysis To do: [CLRS] 4 # T() = O()

More information

Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling

Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed Multi-Evet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria

More information

Handout: How to calculate time complexity? CSE 101 Winter 2014

Handout: How to calculate time complexity? CSE 101 Winter 2014 Hadout: How to calculate time complexity? CSE 101 Witer 014 Recipe (a) Kow algorithm If you are usig a modied versio of a kow algorithm, you ca piggyback your aalysis o the complexity of the origial algorithm

More information

Present Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value

Present 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 information

Estimating the Mean and Variance of a Normal Distribution

Estimating the Mean and Variance of a Normal Distribution Estimatig the Mea ad Variace of a Normal Distributio Learig Objectives After completig this module, the studet will be able to eplai the value of repeatig eperimets eplai the role of the law of large umbers

More information

hp calculators HP 12C Platinum Statistics - correlation coefficient The correlation coefficient HP12C Platinum correlation coefficient

hp calculators HP 12C Platinum Statistics - correlation coefficient The correlation coefficient HP12C Platinum correlation coefficient HP 1C Platium Statistics - correlatio coefficiet The correlatio coefficiet HP1C Platium correlatio coefficiet Practice fidig correlatio coefficiets ad forecastig HP 1C Platium Statistics - correlatio coefficiet

More information

RADICALS AND SOLVING QUADRATIC EQUATIONS

RADICALS AND SOLVING QUADRATIC EQUATIONS RADICALS AND SOLVING QUADRATIC EQUATIONS Evaluate Roots Overview of Objectives, studets should be able to:. Evaluate roots a. Siplify expressios of the for a b. Siplify expressios of the for a. Evaluate

More information

represented by 4! different arrangements of boxes, divide by 4! to get ways

represented by 4! different arrangements of boxes, divide by 4! to get ways Problem Set #6 solutios A juggler colors idetical jugglig balls red, white, ad blue (a I how may ways ca this be doe if each color is used at least oce? Let us preemptively color oe ball i each color,

More information

x : X bar Mean (i.e. Average) of a sample

x : X bar Mean (i.e. Average) of a sample A quick referece for symbols ad formulas covered i COGS14: MEAN OF SAMPLE: x = x i x : X bar Mea (i.e. Average) of a sample x i : X sub i This stads for each idividual value you have i your sample. For

More information

Quadratics - Revenue and Distance

Quadratics - Revenue and Distance 9.10 Quadratics - Reveue ad Distace Objective: Solve reveue ad distace applicatios of quadratic equatios. A commo applicatio of quadratics comes from reveue ad distace problems. Both are set up almost

More information

Section 6.1 Radicals and Rational Exponents

Section 6.1 Radicals and Rational Exponents Sectio 6.1 Radicals ad Ratioal Expoets Defiitio of Square Root The umber b is a square root of a if b The priciple square root of a positive umber is its positive square root ad we deote this root by usig

More information

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means)

CHAPTER 7: Central Limit Theorem: CLT for Averages (Means) CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:

More information

Sum of Exterior Angles of Polygons TEACHER NOTES

Sum of Exterior Angles of Polygons TEACHER NOTES Sum of Exterior Agles of Polygos TEACHER NOTES Math Objectives Studets will determie that the iterior agle of a polygo ad a exterior agle of a polygo form a liear pair (i.e., the two agles are supplemetary).

More information

{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers

{{1}, {2, 4}, {3}} {{1, 3, 4}, {2}} {{1}, {2}, {3, 4}} 5.4 Stirling Numbers . Stirlig Numbers Whe coutig various types of fuctios from., we quicly discovered that eumeratig the umber of oto fuctios was a difficult problem. For a domai of five elemets ad a rage of four elemets,

More information

Trigonometric 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

Trigonometric 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 information

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13

Discrete 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 information

4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then

4 n. n 1. You shold think of the Ratio Test as a generalization of the Geometric Series Test. For example, if a n ar n is a geometric sequence then SECTION 2.6 THE RATIO TEST 79 2.6. THE RATIO TEST We ow kow how to hadle series which we ca itegrate (the Itegral Test), ad series which are similar to geometric or p-series (the Compariso Test), but of

More information

Measures of Spread and Boxplots Discrete Math, Section 9.4

Measures 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 information

MATH /2003. Assignment 4. Due January 8, 2003 Late penalty: 5% for each school day.

MATH /2003. Assignment 4. Due January 8, 2003 Late penalty: 5% for each school day. MATH 260 2002/2003 Assigmet 4 Due Jauary 8, 2003 Late pealty: 5% for each school day. 1. 4.6 #10. A croissat shop has plai croissats, cherry croissats, chocolate croissats, almod croissats, apple croissats

More information

UNIT 3 SUMMARY STATIONS THROUGHOUT THE NEXT 2 DAYS, WE WILL BE SUMMARIZING THE CONCEPT OF EXPONENTIAL FUNCTIONS AND THEIR VARIOUS APPLICATIONS.

UNIT 3 SUMMARY STATIONS THROUGHOUT THE NEXT 2 DAYS, WE WILL BE SUMMARIZING THE CONCEPT OF EXPONENTIAL FUNCTIONS AND THEIR VARIOUS APPLICATIONS. Name: Group Members: UNIT 3 SUMMARY STATIONS THROUGHOUT THE NEXT DAYS, WE WILL BE SUMMARIZING THE CONCEPT OF EXPONENTIAL FUNCTIONS AND THEIR VARIOUS APPLICATIONS. EACH ACTIVITY HAS A COLOR THAT CORRESPONDS

More information

Determining the sample size

Determining 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 information

Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem

Lecture 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 information

Chapter Suppose you wish to use the Principle of Mathematical Induction to prove that 1 1! + 2 2! + 3 3! n n! = (n + 1)! 1 for all n 1.

Chapter Suppose you wish to use the Principle of Mathematical Induction to prove that 1 1! + 2 2! + 3 3! n n! = (n + 1)! 1 for all n 1. Chapter 4. Suppose you wish to prove that the followig is true for all positive itegers by usig the Priciple of Mathematical Iductio: + 3 + 5 +... + ( ) =. (a) Write P() (b) Write P(7) (c) Write P(73)

More information

GCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number.

GCSE 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 information

A probabilistic proof of a binomial identity

A 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 information

where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return

where: T = number of years of cash flow in investment's life n = the year in which the cash flow X n i = IRR = the internal rate of return EVALUATING ALTERNATIVE CAPITAL INVESTMENT PROGRAMS By Ke D. Duft, Extesio Ecoomist I the March 98 issue of this publicatio we reviewed the procedure by which a capital ivestmet project was assessed. The

More information

Counting II 3, 7 3, 2 3, 9 7, 2 7, 9 2, 9

Counting II 3, 7 3, 2 3, 9 7, 2 7, 9 2, 9 Coutig II Sometimes we will wat to choose objects from a set of objects, ad we wo t be iterested i orderig them. For example, if you are leavig for vacatio ad you wat to pac your suitcase with three of

More information

COMP 251 Assignment 2 Solutions

COMP 251 Assignment 2 Solutions COMP 251 Assigmet 2 Solutios Questio 1 Exercise 8.3-4 Treat the umbers as 2-digit umbers i radix. Each digit rages from 0 to 1. Sort these 2-digit umbers ith the RADIX-SORT algorithm preseted i Sectio

More information

CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations

CS103A 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 information

CHAPTER 11 Financial mathematics

CHAPTER 11 Financial mathematics CHAPTER 11 Fiacial mathematics I this chapter you will: Calculate iterest usig the simple iterest formula ( ) Use the simple iterest formula to calculate the pricipal (P) Use the simple iterest formula

More information

8.5 Alternating infinite series

8.5 Alternating infinite series 65 8.5 Alteratig ifiite series I the previous two sectios we cosidered oly series with positive terms. I this sectio we cosider series with both positive ad egative terms which alterate: positive, egative,

More information

Sequences, Series and Convergence with the TI 92. Roger G. Brown Monash University

Sequences, Series and Convergence with the TI 92. Roger G. Brown Monash University Sequeces, Series ad Covergece with the TI 92. Roger G. Brow Moash Uiversity email: rgbrow@deaki.edu.au Itroductio. Studets erollig i calculus at Moash Uiversity, like may other calculus courses, are itroduced

More information

1 Computing the Standard Deviation of Sample Means

1 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 information

Algebra Vocabulary List (Definitions for Middle School Teachers)

Algebra Vocabulary List (Definitions for Middle School Teachers) Algebra Vocabulary List (Defiitios for Middle School Teachers) A Absolute Value Fuctio The absolute value of a real umber x, x is xifx 0 x = xifx < 0 http://www.math.tamu.edu/~stecher/171/f02/absolutevaluefuctio.pdf

More information