SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

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.

Example: Consider the sequence {a i = i} i=1. Starting with i = 1, since a i = i,

Example: Consider the sequence {a i = i} i=1. Starting with i = 1, since a i = i, Sequeces: Defiitio: A sequece is a fuctio whose domai is the set of atural umbers or a subset of the atural umbers. We usually use the symbol a to represet a sequece, where is a atural umber ad a is the

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. Essential Question How can you use a geometric sequence to describe a pattern?

Geometric Sequences. Essential Question How can you use a geometric sequence to describe a pattern? . Geometric Sequeces Essetial Questio How ca you use a geometric sequece to describe a patter? I a geometric sequece, the ratio betwee each pair of cosecutive terms is the same. This ratio is called the

More information

ways to complete the second task after the first is completed, c ways to complete the third task after the first two have been completed,, and c

ways to complete the second task after the first is completed, c ways to complete the third task after the first two have been completed,, and c 9. Basic Combiatorics Pre Calculus 9. BASIC COMBINATORICS Learig Targets:. Solve coutig problems usig tree diagrams, lists, ad/or the multiplicatio coutig priciple 2. Determie whether a situatio is couted

More information

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

4.1 Polynomial Functions and Models

4.1 Polynomial Functions and Models 41 Polyomial Fuctios ad Models Sectio 41 Notes Page 1 Polyomial Fuctio: a x a 1 x a x a 1 1 o The i the formula above is called the degree, ad this is the largest expoet of the polyomial A polyomial ca

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

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

SEQUENCE AND SERIES 1.0 INTRODUCTION. Structure

SEQUENCE AND SERIES 1.0 INTRODUCTION. Structure UNIT 1 SEQUENCE AND SERIES Sequece ad Series Structure 1.0 Itroductio 1.1 Objectives 1. Arithmetic Progressio 1.3 Formula for Sum to Terms of a A.P. 1.4 Geometric Progressio 1.5 Sum to Terms of a G.P.

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

NCERT. , a 3. ,... give the same value, i.e., if a k + 1. ,... if the differences a 2. a k

NCERT. , a 3. ,... give the same value, i.e., if a k + 1. ,... if the differences a 2. a k 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 term a.

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

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

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

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

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

Your grandmother and her financial counselor

Your grandmother and her financial counselor Sectio 10. Arithmetic Sequeces 963 Objectives Sectio 10. Fid the commo differece for a arithmetic sequece. Write s of a arithmetic sequece. Use the formula for the geeral of a arithmetic sequece. Use the

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

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

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

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

First, it is important to recall some key definitions and notations before discussing the individual series tests.

First, it is important to recall some key definitions and notations before discussing the individual series tests. Ifiite Series Tests Coverget or Diverget? First, it is importat to recall some key defiitios ad otatios before discussig the idividual series tests. sequece - a fuctio that takes o real umber values a

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

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

4.4 Monotone Sequences and Cauchy Sequences

4.4 Monotone Sequences and Cauchy Sequences 4.4. MONOTONE SEQUENCES AND CAUCHY SEQUENCES 3 4.4 Mootoe Sequeces ad Cauchy Sequeces 4.4. Mootoe Sequeces The techiques we have studied so far require we kow the limit of a sequece i order to prove the

More information

MATH Testing a Series for Convergence

MATH Testing a Series for Convergence MATH 0 - Testig a Series for Covergece Dr. Philippe B. Laval Keesaw State Uiversity October 4, 008 Abstract This hadout is a summary of the most commoly used tests which are used to determie if a series

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

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

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

V. Adamchik 1. Recursions. Victor Adamchik Fall of 2005

V. Adamchik 1. Recursions. Victor Adamchik Fall of 2005 V. Adamchik Recursios Victor Adamchik Fall of 005 Pla. Solvig Liear Recurreces with Costat Coefficiets. Iteratios. Characteristic equatio.3 Geeral solutio.4 Higher order equatios.5 Multiple roots Solvig

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

Winter Camp 2012 Sequences Alexander Remorov. Sequences. Alexander Remorov

Winter Camp 2012 Sequences Alexander Remorov. Sequences. Alexander Remorov Witer Camp 202 Sequeces Alexader Remorov Sequeces Alexader Remorov alexaderrem@gmail.com Warm-up Problem : Give a positive iteger, cosider a sequece of real umbers a 0, a,..., a defied as a 0 = 2 ad =

More information

1 a a a a a Write a formula for a 6 in terms of a 5.

1 a a a a a Write a formula for a 6 in terms of a 5. Sectio 1.1 Arithmetic Sequeces ad Series Basic Terms A sequece is a ordered list of umbers. Each umber i a sequece is called a term. Work Together Usig the sequece give at the right. Fid the sixth term

More information

ARITHMETIC SEQUENCES

ARITHMETIC SEQUENCES ARITHMETIC SEQUENCES additio subtractio multiplicatio (product) value eve umbers odd umbers sequece term of the sequece arithmetic progressio(sequece) recurrece formula commo differece d cosecutive series

More information

Combining Multiple Averaged Data Points And Their Errors

Combining Multiple Averaged Data Points And Their Errors Combiig Multiple Averaged Data Poits Ad Their Errors Ke Tatebe August 10, 005 It is stadard practice to average measured data poits i order to suppress statistical errors i the fial results. Here, the

More information

addtionalmathematicsprogressionsad dtionalmathematicsprogressionsaddti onalmathematicsprogressionsaddtion almathematicsprogressionsaddtional

addtionalmathematicsprogressionsad dtionalmathematicsprogressionsaddti onalmathematicsprogressionsaddtion almathematicsprogressionsaddtional addtioalmathematicsprogressiosad dtioalmathematicsprogressiosaddti oalmathematicsprogressiosaddtio almathematicsprogressiosaddtioal PROGRESSION mathematicsprogressiosaddtioalma Name thematicsprogressiosaddtioalmathe...

More information

Arithmetic Sequences

Arithmetic Sequences . Arithmetic Sequeces Essetial Questio How ca you use a arithmetic sequece to describe a patter? A arithmetic sequece is a ordered list of umbers i which the differece betwee each pair of cosecutive terms,

More information

Revising algebra skills. Jackie Nicholas

Revising algebra skills. Jackie Nicholas Mathematics Learig Cetre Revisig algebra skills Jackie Nicholas c 005 Uiversity of Sydey Mathematics Learig Cetre, Uiversity of Sydey 1 1 Revisio of Algebraic Skills 1.1 Why use algebra? Algebra is used

More information

5.3 Annuities. Question 1: What is an ordinary annuity? Question 2: What is an annuity due? Question 3: What is a sinking fund?

5.3 Annuities. Question 1: What is an ordinary annuity? Question 2: What is an annuity due? Question 3: What is a sinking fund? 5.3 Auities Questio 1: What is a ordiary auity? Questio : What is a auity due? Questio 3: What is a sikig fud? A sequece of paymets or withdrawals made to or from a accout at regular time itervals is called

More information

Limits and an Introduction to Calculus

Limits and an Introduction to Calculus Chapter Sectio. Itroductio to Limits Limits ad a Itroductio to Calculus Objective: I this lesso ou leared how to estimate limits ad use properties ad operatios of limits. Course Number Istructor Date I.

More information

BINOMIAL THEOREM INTRODUCTION. Development of Binomial Theorem

BINOMIAL THEOREM INTRODUCTION. Development of Binomial Theorem 8 BINOMIAL THEOREM INTRODUCTION I previous classes, you have leart the squares ad cubes of biomial epressios like a + b, a b ad used these to fid the values of umbers like (10), (998) by epressig these

More information

Chapter 11: Sequences and Series

Chapter 11: Sequences and Series Algebra II Hoors Chapter : Sequeces ad Series Sequeces ad Summatio Notatio A sequece is a ordered list of umbers; for example,, /4, 5/9, 7/6, The graph of a sequece is a series of ucoected poits with domai,,,

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

Sequences and Series Binomial Theorem

Sequences and Series Binomial Theorem Mrs. Turer s Precalculus page 0 Sequeces ad Series Biomial Theorem This is the Golde Spiral a special ratio that occurs i ature Name: Period: Mrs. Turer s Precalculus page 8. Sequeces ad Series Notes What

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

Now that we know what our sample looks like, we d like to be able to describe it numerically.

Now that we know what our sample looks like, we d like to be able to describe it numerically. Descriptive Statistics: Now that we kow what our sample looks like, we d like to be able to describe it umerically. I other words, istead of havig a lot of umbers (oe for each record), we d like to be

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

MISCELLANEOUS SEQUENCES & SERIES QUESTIONS

MISCELLANEOUS SEQUENCES & SERIES QUESTIONS MISCELLANEOUS SEQUENCES & SERIES QUESTIONS Questio (***+) Evaluate the followig sum 30 r ( 2) 4r 78. 3 MP2-V, 75,822,200 Questio 2 (***+) Three umbers, A, B, C i that order, are i geometric progressio

More 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

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

Steve Smith Tuition: Maths Notes

Steve Smith Tuition: Maths Notes Steve Smith Tuitio: Maths Notes e iπ + 1 = 0 a 2 + b 2 = c 2 z +1 = z 2 + c V E + F = 2 Fidig Equatios of Regressio Lies Cotets 1 Itroductio 3 1.1 A Quick Overview of Correlatio.............................

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

Intro to Sequences / Arithmetic Sequences and Series Levels

Intro to Sequences / Arithmetic Sequences and Series Levels Itro to Sequeces / Arithmetic Sequeces ad Series Levels Level : pg. 569: #7, 0, 33 Pg. 575: #, 7, 8 Pg. 584: #8, 9, 34, 36 Levels, 3, ad 4(Fiboacci Sequece Extesio) See Hadout Check for Uderstadig Level

More information

SECTION 8-4 Binomial Formula

SECTION 8-4 Binomial Formula 8- Biomial Formula 581 poit of view it appears that he will ever fiish the race. This famous paradox is attributed to the Gree philosopher Zeo, 95 35 B.C. If we assume the ruer rus at 0 yards per miute,

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

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

Section 1.3 Convergence Tests

Section 1.3 Convergence Tests Sectio.3 Covergece Tests We leared i the previous sectio that for ay geometric series, we kow exactly whether it coverges ad the value it coverges to. For most series, however, there is o exact formula

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

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

SEQUENCE AND SERIES. Chapter Overview

SEQUENCE AND SERIES. Chapter Overview Chapter 9 SEQUENCE AND SERIES 9.1 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 1, a,..., etc., the subscript deotes

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

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

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

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

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

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

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

2.5. THE COMPARISON TEST

2.5. THE COMPARISON TEST SECTION. THE COMPARISON TEST 69.. THE COMPARISON TEST We bega our systematic study of series with geometric series, provig the Geometric Series Test: ar coverges if ad oly if r. The i the last sectio we

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

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

An Example of ASM Design: A Binary Multiplier

An Example of ASM Design: A Binary Multiplier A Example of ASM Desig: A Biary Multiplier Itroductio Dr. D. Capso Electrical ad Computer Egieerig McMaster Uiversity A algorithmic state machie (ASM) is a Fiite State Machie that uses a sequetial circuit

More information

Chapter Eleven. Taylor Series. (x a) k. c k. k= 0

Chapter Eleven. Taylor Series. (x a) k. c k. k= 0 Chapter Eleve Taylor Series 111 Power Series Now that we are kowledgeable about series, we ca retur to the problem of ivestigatig the approximatio of fuctios by Taylor polyomials of higher ad higher degree

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

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

The Binomial Theorem

The Binomial Theorem The Biomial Theorem Itroductio You should be familiar with the followig formula: x + y 2 = x 2 + 2xy + y 2 The biomial theorem explais how to get a correspodig expasio whe the expoet is a arbitrary atural

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

Sequences and Series. Growth of $5,000 investment per year. Amount (in thousands of dollars) 150. Time (years)

Sequences and Series. Growth of $5,000 investment per year. Amount (in thousands of dollars) 150. Time (years) C H A P T E R Sequeces ad Series Amout (i thousads of dollars) 50 00 50 0 Growth of $5,000 ivestmet per year 3 4 5 6 7 8 9 0 Time (years) veryoe realizes the importace of ivestig for the future. Some people

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

Randomized Algorithms I, Spring 2016, Department of Computer Science, University of Helsinki Homework 1: Solutions (Discussed January 29, 2016)

Randomized Algorithms I, Spring 2016, Department of Computer Science, University of Helsinki Homework 1: Solutions (Discussed January 29, 2016) Radomized Algorithms I, Sprig 0, Departmet of Computer Sciece, Uiversity of Helsiki Homework : Solutios Discussed Jauary 9, 0). Exercise.: Cosider the followig balls-ad-bi game. We start with oe black

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

How many prime numbers are there?

How many prime numbers are there? CHAPTER I: PRIME NUMBERS Sectio : Types of Primes I the previous sectios, we saw the importace of prime umbers. But how may prime umbers are there? How may primes are there less tha a give umber? Do they

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

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

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

Combinatorics Dan Hefetz, Richard Mycroft (lecturer)

Combinatorics Dan Hefetz, Richard Mycroft (lecturer) Combiatorics Da Hefetz, Richard Mycroft (lecturer) These are otes of the lectures give by Richard Mycroft for the course Commuicatio Theory ad Combiatorics (3P16/4P16) at the Uiversity of Birmigham, o

More information

Mgr. ubomíra Tomková. Limit

Mgr. ubomíra Tomková. Limit Limit I mathematics, the cocept of a "limit" is used to describe the behaviour of a fuctio as its argumet either gets "close" to some poit, or as it becomes arbitrarily large; or the behaviour of a sequece's

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

Solutions to Problem Set 0: Math 149S

Solutions to Problem Set 0: Math 149S Solutios to Problem Set 0: Math 49S Matthew Roglie September 8, 009 Some terms or cocepts here may be ufamiliar to you. We will cover them later, but i the meatime you are strogly ecouraged to google them!

More information

Algorithms, Design and Analysis. Prefix Averages (Quadratic) Computing Prefix Averages. Prefix Averages (Linear, nonrecursive)

Algorithms, Design and Analysis. Prefix Averages (Quadratic) Computing Prefix Averages. Prefix Averages (Linear, nonrecursive) Algorithms, Desig ad Aalysis Itroductio. Algorithm A algorithm is a sequece of uambiguous istructios for solvig a problem, i.e., for obtaiig a required output for ay legitimate iput i a fiite amout of

More information

Section 1.1 1, 1 2, 1 4, 1 8,...

Section 1.1 1, 1 2, 1 4, 1 8,... Differece Equatios to Differetial Equatios Sectio. Calculus: Areas Ad Tagets The study of calculus begis with questios about chage. What happes to the velocity of a swigig pedulum as its positio chages?

More information

The Discrete Fourier Transform

The Discrete Fourier Transform The Discrete Fourier Trasform Fracis J. Narcowich October 4, 2005 1 Motivatio We wat to umerically approximate coefficiets i a Fourier series. The first step is to see how the trapezoidal rule applies

More information

Math 129 Midterm 3 Study Guide

Math 129 Midterm 3 Study Guide Math 29 Midterm Study Guide As is usually the case, this is ot iteded to be comprehesive but merely to serve as a guide to help make your study time more e ciet. Brief Review Sectio 8.5 Work will be covered

More information

3.3 Convergence Tests for Infinite Series

3.3 Convergence Tests for Infinite Series 3.3 Covergece Tests for Ifiite Series 3.3. The itegral test We may plot the sequece a i the Cartesia plae, with idepedet variable ad depedet variable a. The sum a ca the be represeted geometrically as

More information

The Integral Test. and. n 1. either both converge or both diverge. f i f 2 f 3... f n. f i f 1 f 2... f n 1. f i f x dx n 1. f i. i 1.

The Integral Test. and. n 1. either both converge or both diverge. f i f 2 f 3... f n. f i f 1 f 2... f n 1. f i f x dx n 1. f i. i 1. 0_090.qd //0 :5 PM Page 7 SECTIO 9. The Itegral Test ad p-series 7 Sectio 9. The Itegral Test ad p-series Use the Itegral Test to determie whether a ifiite series coverges or diverges. Use properties of

More information

2. Introduction to Statistics and Sampling

2. Introduction to Statistics and Sampling 2. Itroductio to Statistics ad Samplig 2. Itroductio 2.. Defiitio of statistics, statistic 2.2.2 Why study Statistics? 2.2 Defiitios 2.2. Populatio 2.2.2 Sample 2.2.3 Probability 2.2.4 Cotiuous vs. discrete

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

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