8.1 Arithmetic Sequences

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "8.1 Arithmetic Sequences"

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?

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

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

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

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

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

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

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

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

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

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

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

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

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

.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

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

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

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

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

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

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

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

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

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

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

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

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

A function f whose domain is the set of positive integers is called a sequence. The values

A function f whose domain is the set of positive integers is called a sequence. The values EQUENCE: A fuctio f whose domi is the set of positive itegers is clled sequece The vlues f ( ), f (), f (),, f (), re clled the terms of the sequece; f() is the first term, f() is the secod term, f() is

More information

SEQUENCES AND SERIES CHAPTER

SEQUENCES 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 information

SEQUENCES AND SERIES

SEQUENCES 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 information

One-step equations. Vocabulary

One-step equations. Vocabulary Review solvig oe-step equatios with itegers, fractios, ad decimals. Oe-step equatios Vocabulary equatio solve solutio iverse operatio isolate the variable Additio Property of Equality Subtractio Property

More information

Math : Sequences and Series

Math : Sequences and Series EP-Program - Strisuksa School - Roi-et Math : Sequeces ad Series Dr.Wattaa Toutip - Departmet of Mathematics Kho Kae Uiversity 00 :Wattaa Toutip wattou@kku.ac.th http://home.kku.ac.th/wattou. Sequeces

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

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

Essential Question How can you use properties of exponents to simplify products and quotients of radicals?

Essential Question How can you use properties of exponents to simplify products and quotients of radicals? . Properties of Ratioal Expoets ad Radicals Essetial Questio How ca you use properties of expoets to simplify products ad quotiets of radicals? Reviewig Properties of Expoets Work with a parter. Let a

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

NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL 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 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

Arithmetic Sequences

Arithmetic Sequences Arithmetic equeces A simple wy to geerte sequece is to strt with umber, d dd to it fixed costt d, over d over gi. This type of sequece is clled rithmetic sequece. Defiitio: A rithmetic sequece is sequece

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

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

2.7 Sequences, Sequences of Sets

2.7 Sequences, Sequences of Sets 2.7. SEQUENCES, SEQUENCES OF SETS 67 2.7 Sequeces, Sequeces of Sets 2.7.1 Sequeces Defiitio 190 (sequece Let S be some set. 1. A sequece i S is a fuctio f : K S where K = { N : 0 for some 0 N}. 2. For

More information

Sample. Activity Library: Volume II. Activity Collections. Featuring real-world context collections:

Sample. Activity Library: Volume II. Activity Collections. Featuring real-world context collections: Activity Library: Volume II Sample Activity Collectios Featurig real-world cotext collectios: Arithmetic II Fractios, Percets, Decimals III Fractios, Percets, Decimals IV Geometry I Geometry II Graphig

More information

Unit 2 Sequences and Series

Unit 2 Sequences and Series Accelerated Mathematics III Frameworks Studet Editio Uit Sequeces ad Series d Editio April, 011 Table of Cotets INTRODUCTION:... 3 Reaissace Festival Learig Task... 8 Fasciatig Fractals Learig Task...

More information

Unit 2 Sequences and Series

Unit 2 Sequences and Series Mathematics IV Uit 1 st Editio Mathematics IV Frameworks Studet Editio Uit Sequeces ad Series 1 st Editio Kathy Cox, State Superitedet of Schools Uit : Page 1 of 35 Mathematics IV Uit 1 st Editio Table

More information

FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix

FIBONACCI 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 information

MATHEMATICS P1 COMMON TEST JUNE 2014 NATIONAL SENIOR CERTIFICATE GRADE 12

MATHEMATICS 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 information

8.3 POLAR FORM AND DEMOIVRE S THEOREM

8.3 POLAR FORM AND DEMOIVRE S THEOREM SECTION 8. POLAR FORM AND DEMOIVRE S THEOREM 48 8. POLAR FORM AND DEMOIVRE S THEOREM Figure 8.6 (a, b) b r a 0 θ Complex Number: a + bi Rectagular Form: (a, b) Polar Form: (r, θ) At this poit you ca add,

More information

7 b) 0. Guided Notes for lesson P.2 Properties of Exponents. If a, b, x, y and a, b, 0, and m, n Z then the following properties hold: 1 n b

7 b) 0. Guided Notes for lesson P.2 Properties of Exponents. If a, b, x, y and a, b, 0, and m, n Z then the following properties hold: 1 n b Guided Notes for lesso P. Properties of Expoets If a, b, x, y ad a, b, 0, ad m, Z the the followig properties hold:. Negative Expoet Rule: b ad b b b Aswers must ever cotai egative expoets. Examples: 5

More information

Unit 8 Rational Functions

Unit 8 Rational Functions Uit 8 Ratioal Fuctios Algebraic Fractios: Simplifyig Algebraic Fractios: To simplify a algebraic fractio meas to reduce it to lowest terms. This is doe by dividig out the commo factors i the umerator ad

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

Department of Computer Science, University of Otago

Department 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 information

Continued Fractions continued. 3. Best rational approximations

Continued Fractions continued. 3. Best rational approximations Cotiued Fractios cotiued 3. Best ratioal approximatios We hear so much about π beig approximated by 22/7 because o other ratioal umber with deomiator < 7 is closer to π. Evetually 22/7 is defeated by 333/06

More information

CHAPTER 19 NUMBER SEQUENCES

CHAPTER 19 NUMBER SEQUENCES EXERCISE 77 Page 167 CHAPTER 19 NUMBER SEQUENCES 1. Determie the ext two terms i the series: 5, 9, 13, 17, It is oticed that the sequece 5, 9, 13, 17,... progressively icreases by 4, thus the ext two terms

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

Tommy R. Jensen, Department of Mathematics, KNU. 1 Number Sequences 1. 2 Special Number Sequences 2. 3 Generating Functions 4

Tommy R. Jensen, Department of Mathematics, KNU. 1 Number Sequences 1. 2 Special Number Sequences 2. 3 Generating Functions 4 Part 13 Geeratig Fuctios Prited versio of the lecture Discrete Mathematics o 1 October 009 Tommy R Jese, Departmet of Mathematics, KNU 131 Cotets 1 Number Sequeces 1 Special Number Sequeces 3 Geeratig

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

SEQUENCES AND SERIES. Chapter Nine

SEQUENCES AND SERIES. Chapter Nine Chapter Nie SEQUENCES AND SERIES I this chapter, we look at ifiite lists of umbers, called sequeces, ad ifiite sums, called series. I Sectio 9., we study sequeces. I Sectio 9.2, we begi with a particular

More information

9.1 Simplify Radical Expressions

9.1 Simplify Radical Expressions 9.1 Simplifyig Radical Expressios (Page 1 of 20) 9.1 Simplify Radical Expressios Radical Notatio for the -th Root of a If is a iteger greater tha oe, the the th root of a is the umer whose th power is

More information

Grade 7. Strand: Number Specific Learning Outcomes It is expected that students will:

Grade 7. Strand: Number Specific Learning Outcomes It is expected that students will: Strad: Number Specific Learig Outcomes It is expected that studets will: 7.N.1. Determie ad explai why a umber is divisible by 2, 3, 4, 5, 6, 8, 9, or 10, ad why a umber caot be divided by 0. [C, R] [C]

More information

Math Discrete Math Combinatorics MULTIPLICATION PRINCIPLE:

Math Discrete Math Combinatorics MULTIPLICATION PRINCIPLE: Math 355 - Discrete Math 4.1-4.4 Combiatorics Notes MULTIPLICATION PRINCIPLE: If there m ways to do somethig ad ways to do aother thig the there are m ways to do both. I the laguage of set theory: Let

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

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

1 Correlation and Regression Analysis

1 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 information

3.2 Introduction to Infinite Series

3.2 Introduction to Infinite Series 3.2 Itroductio to Ifiite Series May of our ifiite sequeces, for the remaider of the course, will be defied by sums. For example, the sequece S m := 2. () is defied by a sum. Its terms (partial sums) are

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

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

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

NATIONAL SENIOR CERTIFICATE GRADE 11

NATIONAL 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 information

Your 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:

Your 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 information

Binet Formulas for Recursive Integer Sequences

Binet Formulas for Recursive Integer Sequences Biet Formulas for Recursive Iteger Sequeces Homer W. Austi Jatha W. Austi Abstract May iteger sequeces are recursive sequeces ad ca be defied either recursively or explicitly by use of Biet-type formulas.

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

MESSAGE TO TEACHERS: NOTE TO EDUCATORS:

MESSAGE TO TEACHERS: NOTE TO EDUCATORS: MESSAGE TO TEACHERS: NOTE TO EDUCATORS: Attached herewith, please fid suggested lesso plas for term 1 of MATHEMATICS Grade 12. Please ote that these lesso plas are to be used oly as a guide ad teachers

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

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

{{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

FOUNDATIONS OF MATHEMATICS AND PRE-CALCULUS GRADE 10

FOUNDATIONS 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 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

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

Chapter 7. In the questions below, describe each sequence recursively. Include initial conditions and assume that the sequences begin with a 1.

Chapter 7. In the questions below, describe each sequence recursively. Include initial conditions and assume that the sequences begin with a 1. Use the followig to aswer questios -6: Chapter 7 I the questios below, describe each sequece recursively Iclude iitial coditios ad assume that the sequeces begi with a a = 5 As: a = 5a,a = 5 The Fiboacci

More information

A black- line master of Example 3 You Try is on provided on page 10 for duplication or use with a projection system.

A black- line master of Example 3 You Try is on provided on page 10 for duplication or use with a projection system. Grde Level/Course: Algebr Lesso/Uit Pl Nme: Geometric Sequeces Rtiole/Lesso Abstrct: Wht mkes sequece geometric? This chrcteristic is ddressed i the defiitio of geometric sequece d will help derive the

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

1. a n = 2. a n = 3. a n = 4. a n = 5. a n = 6. a n =

1. a n = 2. a n = 3. a n = 4. a n = 5. a n = 6. a n = Versio PREVIEW Homework Berg (5860 This prit-out should have 9 questios. Multiple-choice questios may cotiue o the ext colum or page fid all choices before aswerig. CalCb0b 00 0.0 poits Rewrite the fiite

More information

G r a d e. 5 M a t h e M a t i c s. Patterns and relations

G r a d e. 5 M a t h e M a t i c s. Patterns and relations G r a d e 5 M a t h e M a t i c s Patters ad relatios Grade 5: Patters ad Relatios (Patters) (5.PR.1) Edurig Uderstadigs: Number patters ad relatioships ca be represeted usig variables. Geeral Outcome:

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

Mathematicians have been fascinated by the majestic simplicity of the Fibonacci

Mathematicians have been fascinated by the majestic simplicity of the Fibonacci Joh Holde Tutoa3000@aol.com Ivertig the iboacci Sequece Mathematicias have bee fasciated by the majestic simplicity of the iboacci Sequece for ceturies. It starts as a simple,,, 3, 5, 8,3,... computed

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

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

Lecture Notes CMSC 251

Lecture Notes CMSC 251 We have this messy summatio to solve though First observe that the value remais costat throughout the sum, ad so we ca pull it out frot Also ote that we ca write 3 i / i ad (3/) i T () = log 3 (log ) 1

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

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

SENIOR CERTIFICATE EXAMINATIONS

SENIOR CERTIFICATE EXAMINATIONS SENIOR CERTIFICATE EXAMINATIONS MATHEMATICS P1 016 MARKS: 150 TIME: 3 hours This questio paper cosists of 9 pages ad 1 iformatio sheet. Please tur over Mathematics/P1 DBE/016 INSTRUCTIONS AND INFORMATION

More information

Properties of MLE: consistency, asymptotic normality. Fisher information.

Properties 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 information

Mathematical goals. Starting points. Materials required. Time needed

Mathematical 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 information

a 1 1 a 2 1 a a n 1 ò a n or o a n n n 1 32, 63 16, 31

a 1 1 a 2 1 a a n 1 ò a n or o a n n n 1 32, 63 16, 31 206 Cegage Learig. All Rights Reserved. This cotet is ot yet fial ad Cegage Learig Series The curret record for computig a decimal approximatio for was obtaied by Shigeru Kodo ad Alexader Yee i 20 ad cotais

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

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

3. Greatest Common Divisor - Least Common Multiple

3. 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 information

NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE 12 NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P NOVEMBER 009() MARKS: 50 TIME: 3 hours This questio paper cosists of 0 pages, a iformatio sheet ad diagram sheet. Please tur over Mathematics/P DoE/November

More information