SEQUENCES AND SERIES


 Douglas Barrie Robinson
 1 years ago
 Views:
Transcription
1 9 SEQUENCES AND SERIES INTRODUCTION Sequeces have may importat applicatios i several spheres of huma activities Whe a collectio of objects is arraged i a defiite order such that it has a idetified first member, secod member, third member ad so o, we say that the collectio is listed i a sequece For example : (i) The amout of moey i a fixed deposit i a bak over a umber of years occur i a sequece (ii) Depreciated values of certai commodities occur i a sequece (iii) The populatio of bacteria at differet times forms a sequece (iv) Cosider the umber of acestors ie parets, gradparets, great gradparets etc that a perso has over 0 geeratios The umbers of perso s acestors for the first, secod, third,, teth geeratios are,, 8,, 0 These umbers form a sequece Sequeces, followig specific patters are called progressios I the previous class, you have already studied about arithmetic progressio (AP) I this chapter, besides studyig more about AP, we shall also study arithmetic mea (AM), geometric progressio (GP), geometric mea (GM), relatioship betwee AM ad GM, arithmeticogeometric series ad sum to terms of special series Σ, Σ, Σ etc 9 SEQUENCE A set of umbers arraged i a defiite order accordig to some defiite rule (or rules) is called a sequece Each umber of the set is called a term of the sequece A sequece is called fiite or ifiite accordig as the umber of terms i it is fiite or ifiite The differet terms of a sequece are usually deoted by a, a, or by T, T, T, The subscript (always a atural umber) deotes the positio of the term i the sequece The umber occurrig at the th place of a sequece ie a is called the geeral term of the sequece A fiite sequece is described by a, a,, a ad a ifiite sequece is described by a, a, to If all the terms are real, we have a real sequece; if all terms are complex umbers, we have a complex sequece etc For example, cosider the followig sequeces : (i), 5, 7, 9,, (ii) 8, 5,,,, (iii), 6, 8, 5,, 58 (iv), 8 (v),, 9, 6, (vi),, 5, 7,,, (vii),,,, 5, 8,,
2 98 MATHEMATICS XI We observe the followig : (i) Here each term is obtaied by addig to the previous term (ii) Here each term is obtaied by subtractig from the precedig term (iii) Here each term is obtaied by multiplyig the precedig term by (iv) Here each term is obtaied by multiplyig the precedig term by (v) Here each term is obtaied by squarig the ext atural umber (vi) This is the sequece of prime umbers (vii) Here each term after secod term is obtaied by addig the previous two terms Also ote that sequeces (i) ad (iii) are fiite sequeces whereas others are ifiite sequeces Moreover to defie a sequece, we eed ot always have a explicit formula for the th term Till today, obody has foud the formula for th prime umber Also ote that, i (i) a = a + (ii) a = a (iii) a = a (iv) a = a (v) a = (vii) a = a + a ( > ) ad i (vi) we may describe a = th prime umber If the terms of a sequece ca be described by a explicit formula, the the sequece is called a progressio Note that the sequeces (i) to (v) give above are all progressios, whereas sequece (vi) is ot a progressio The sequece (vii) ie,,,, 5, 8,,, is also a progressio It is called Fiboacci sequece 9 SERIES If the terms of a sequece are coected by plus sigs we get a series Thus, if a, a,, is a give sequece the the expressio a + a + is called the series associated with the give sequece The series is fiite or ifiite accordig as the give sequece is fiite or ifiite From sequeces (i) to (v) give above, we ca form followig series : (i) (ii) ( ) + ( ) + (iii) (iv) (v) If a deotes the geeral term of a sequece, the a + a + + a is a series of terms I a series a + a + + a k +, the sum of first terms is deoted by S Thus S = a + a + + a = a k k = If S deotes the sum of terms of a sequece, the S S = (a + a + + a ) (a + a + + a ) = a Thus, a = S S REMARK The word series is referred to the idicated sum ot to the sum itself For example, is a fiite series with five terms By the words sum of a series will mea the umber that results from addig the terms, so the sum of the above series is 5
3 SEQUENCES AND SERIES 99 ILLUSTRATIVE EXAMPLES Example Fid the ext term of the sequece (i),, 6, 8 (ii),,, (iii), 8,, 8 (iv),, 5, 7 (v), 8, 7, 6 Solutio (i) We see that each term is obtaied by addig to the previous term Hece, the ext term = 8 + = 0 (ii) Here we see that each term is obtaied by multiplyig the previous term by Hece, the ext term = = 8 (iii) We see that each term is obtaied by multiplyig the previous term by Hece, the ext term = 8 = 5 (iv) Here each term is obtaied by subtractig from the previous term Hece, the ext term = 7 = 9 (v) We see that terms are cubes of atural umbers,,, Hece, the ext term of the sequece is 5 ie 5 Example Write the first four terms of the sequece defied by (i) a = + (ii) a = th prime umber Solutio (i) Give a = + Puttig =,,,, we get a = + = 7, a = + = 9, a = + = 9, a = + = 67 Hece, the first four terms of the give sequece are 7, 9, 9, 67 (ii) We kow that the first four prime umbers are,, 5, 7 Hece, the first four terms of the sequece are,, 5, 7 Example Fid the 0th term of the sequece defied by a = ( ) + Solutio Give a = ( ), puttig = 0, we get + 0 ( 0 ) a 0 = = = 0 + Example Fid the th ad th terms of the sequece defied by a =, whe is eve +, whe is odd Solutio As is odd = + = + = 69 + = 70; ad as is eve, a = = = 96 Example 5 Fid the first five terms of the sequece give by a =, a = + a ad a = a +5 for > Also write the correspodig series Solutio Here a =, a = + a = + = 5 Give a = a + 5 for >, puttig =,, 5, we get a = a + 5 = = 5 a = a + 5 = = 5 a 5 = a + 5 = = 75 Hece, the first five terms of the give sequece are, 5, 5, 5, 75 The correspodig series is
4 00 MATHEMATICS XI Example 6 The Fiboacci sequece is defied by a = a =, a = a + a for >, fid a a for =,,,, 5 Solutio Give a = a = ad a = a + a for > Puttig =,, 5 ad 6 i (i), we get a = a + a = + =, a = a + a = + =, a 5 = a = + = 5 ad a 6 = a 5 + a = 5 + = 8 Puttig =,,, ad 5 i a +, we get a ie a a a a5 a6,,,, a a a a a5, 5, 8,, 5 ie,, 5, 8, 5 + Example 7 (i) Fid the first terms of the series Σ ( ) + (ii) The sum of terms of a series is + for all values of Fid the first terms of the series Also fid its 0th term Solutio (i) th term of the give series a = ( ) + First term = a = ( ) + = (i) Secod term = a = ( ) + = 9 Third term = a = ( ) + = 7 Hece, first terms of the give series are,, 9 7 (ii) Give S = + S = ( ) + ( ) = + a = S S = + ( + ) = + Puttig =,, ad 0, we get a = + =, a = + = 6 = + = 8 ad a 0 = 0 + = Hece, the first three terms are, 6, 8 ad the 0th term is Example 8 If for a sequece, S = ( ), fid its first four terms Solutio Give S = ( ) S = ( ) a = S S = ( ) ( ) = ( ) = ( ) = Puttig =,,,, we get a = 0 =, a = = = = 6 ad a = = 08 Hece, the first four terms of the sequece are,, 6, 08 Example 9 (i) Write (k + ) i expaded form k= (ii) Write the series i sigma otatio +
5 SEQUENCES AND SERIES 0 Solutio (i) Puttig k =,,,,, i (k + ), we get, 5, 0, 7,, + Hece, (k + ) = ( + ) k= (ii) We see that kth term of series = k k + Hece, the give series ca be writte as k = + k + EXERCISE 9 Very short aswer type questios ( to 7) : Give a example of a sequece which is ot a progressio Which term of the sequece give by a = + +, N, is 6? Write the first three terms of the sequece whose th term is give by a = ( ) 5 If a sequece is give by a =, a = + a ad a = a for > The write the correspodig series upto terms 5 Write the ext term of each of the followig sequeces : (i),,,, (ii) 5, 5, 5, 5 8, 6 Write the ext term of each of the followig sequeces : (i) 0,, 6,, 0, (ii) 6, 9, 6, 7,, (iii), 5,, 0, 55, Hit (iii) 5 = +, = 5 +, 0 = +, 7 Write the eleveth term of the followig sequece :,,,, 5, 8,,,, 8 Write first 5 terms of the followig sequeces whose th terms are give by : (i) a = + 5 (ii) a = 6 (iii) a = ( ) (iv) a = ( + ) (v) a = (vi) a = + (vii) a = ( ) 5 + (viii) a = ( + 5 ) (ix) a = + 9 Fid the idicated term(s) i each of the followig sequeces whose th terms are : (i) a = ; a 7, a (ii) a = ; a 5, a 7 (iii) a = ( ) ; a 9 (iv) a = ( ) ( ) ( + ); a, a, a 0 0 Fid the 8th ad 5th terms of the sequece defied by ( + ), if is eve atural umber T =, if is odd atural umber + Fid the first five terms of each of the followig sequeces ad obtai the correspodig series : (i) a =, a = a +, (ii) a =, a = a +, for all > (iii) a =, a = a for (iv) a = a =, a = a for > If the sum of terms of a sequece is give by S = + for all N, fid the first terms Also fid its 0th term k =
6 0 MATHEMATICS XI First term of a sequece is ad the ( + )th term is obtaied by addig ( + ) to the th term for all atural umbers Fid the sixth term of the sequece Hit a + = a + ( + ) for all atural umbers 9 ARITHMETIC PROGRESSION (AP) A sequece ( fiite or ifiite) is called a arithmetic progressio (abbreviated AP) iff the differece of ay term from its precedig term is costat This costat is usually deoted by d ad is called commo differece Thus a, a,, a or a, a, is a AP iff a k + a k = d, a costat (idepedet of k) for k =,,, or k =,,, as the case may be It follows that, i a AP, a + = a + d ie ay term (except the first) is obtaied by addig the fixed umber d to its precedig term If the sequece a, a,, a is a AP, the the series a + a + + a is called a arithmetic series Geeral term of a AP Let a be the first term ad d be the commo differece of a AP, the the AP is a, a + d, a + d, ad its th term = a + ( ) d Hece, geeral term a = a + ( ) d Last term of a AP If the last term of a AP cosistig of terms is deoted by l, the l = a + ( ) d The th term from the ed of a fiite AP (i) If a, a + d, a + d, is a fiite AP cosistig of m terms, the the th term from the ed = (m + )th term from begiig = a + ( m + ) d = a + (m ) d (ii) If a, a + d, a + d, is a fiite AP with last term l, the the th term from ed = l + ( ) ( d) = l ( ) d For, whe we look at the terms of the give AP from the last ad move towards begiig we fid that the sequece is a AP with commo differece d ad first term as l, therefore, th term from the ed of the give AP = l +( ) ( d) = l ( ) d Sum of terms of a AP Let a be the first term, d the commo differece ad l the last term of a AP If S deotes the sum of its first terms, the S = (a + ( ) d) or S = (a + l) REMARKS If the sum of terms of a AP is deoted by S, the its commo differece = S S Three umbers a, b ad c are i AP iff b a = c b ie iff b = a + c Some problems ivolve, or 5 umbers i AP If the sum of the umbers is give, the i a AP, (i) three umbers are take as a d, a, a + d (ii) four umbers are take as a d, a d, a + d, a + d (iii) five umbers are take as a d, a d, a, a + d, a + d This cosiderably simplifies the calculatios of a ad d The sum of the terms equidistat from the begiig ad the ed of a AP is always the same ad equals to the sum of the first ad the last terms
7 SEQUENCES AND SERIES 0 9 If the terms of a arithmetic progressio (AP) are icreased, decreased, multiplied or divided by the same ozero costat, they remai i arithmetic progressio Proof Cosider the AP a, a + d, a + d, a + d, () (i) If each term of () is icreased by a costat k, we obtai the sequece a + k, a + d + k, a + d + k, a + d + k, ie a + k, a + k + d, a + k + d, a + k +d, which is clearly a AP whose first term is a + k ad commo differece is d (ii) If each term of () is decreased by a costat k, we obtai the sequece a k, a + d k, a + d k, ie a k, a k + d, a k + d, a k + d, which is clearly a AP whose first term is a k ad commo differece is d (iii) If each term of () is multiplied by a costat k, we obtai the sequece ak, (a + d) k, (a + d) k, ie ak, ak + dk, ak + dk, ak + dk which is clearly a AP with first term ak ad commo differece dk (iv) Whe each term of () is divided by k ( 0), we obtai the sequece a a d a d a d, +, +, +, k k k k k k k which is clearly a AP with first term a k ad commo differece d k ILLUSTRATIVE EXAMPLES Example The fourth term of AP is equal to times its first term ad seveth term exceeds twice the third term by Fid the first term ad the commo differece Solutio Let a be the first term ad d be the commo differece Now a = a a + d = a d = a (i) a 7 = a + a + 6 d = (a + d) + d = a + (ii) Solvig (i) ad (ii) simultaeously, we get a =, d = Hece the first term of the give sequece is ad commo differece is Example I a sequece, the th term is a = + Show that it is ot a AP Solutio Give a = + a + = ( + ) + a + a = ( ( + ) + ) ( + ) = ( ( + + ) + ) ( + ) Hece the give sequece is ot a AP = +, which depeds upo ad is ot costat Example I a sequece, the sum of first terms is S = P + ( ) Q where P, Q are costats Show that the sequece is a AP Fid its first term, commo differece ad 00th term Solutio Give S = P + ( ) Q, S = ( )P + ( ) ( ) Q a = S S = ( ( ))P + [ ( ) ( ) ( )]Q = P + ( ) ( ( ))Q = P + ( ) Q = P + ( )Q a + = P + Q a + a = (P + Q) (P + ( ) Q) = Q, which is costat Hece the give sequece is a AP with commo differece Q a = P ad a 00 = P + 99Q
8 0 MATHEMATICS XI Example Which term of the sequece 5,,,, is the first egative term? Solutio The give sequece is a AP with commo differece d = a = 5 Let th term of the give AP be the first egative term, the a < ( ) < 0 0 < 0 0 < 0 0 < > 0 > 0 ie > ad first term Sice 5 is the least atural umber satisfyig > = 5 Hece, 5th term of the give sequece is the first egative term Example 5 Which term of the sequece + 8 i, i, i, is (i) real (ii) purely imagiary? Solutio The give sequece is a AP with commo differece d = i ad first term a = + 8 i a (geeral term) = a + ( ) d = ( + 8 i) + ( ) ( i) = ( + ) + (8 + ) i = ( ) + (9 ) i (i) Let th term of the give sequece be real ( ) + (9 ) i is real 9 = 0 = 9 Hece, 9th term of the give sequece is real (ii) Let th term of the give sequece be purely imagiary ( ) + (9 ) i is purely imagiary = 0 = 7 Hece, 7th term of the give sequece is purely imagiary Example 6 How may terms are idetical i the two Arithmetic progressios,, 6, 8, upto 00 terms ad, 6, 9,, upto 80 terms Solutio 00th term of the AP,, 6, 8, = + (00 ) = 00 ad 80th term of the AP, 6, 9,, = + (80 ) = 0 Let terms be idetical i the two give Arithmetic progressios The sequece of idetical terms is 6,, 8, which is a AP with first term 6 ad commo differece 6 Its th term = 6 + ( ) 6 = 6 Sice the last term ie th term of the sequece of idetical terms ca t be greater tha 00, Hece, terms are idetical = NOTE If fially we get m, the = m if m is a iteger ad = a iteger just less tha m if m is ot a iteger
9 8 MATHEMATICS XI EXERCISE 9 ANSWERS,, 5, 7,,, 7, 8th 5, 5, (i) 5 (ii) 6 6 (i) 0 (ii) 6 (iii) (i) 7, 9,,, 5 (ii) 6, 5 6, 7, 6, 6 (iii) 0,, 6,, 0 (iv), 8, 5,, 5 (v),, 8, 6, (vi),, 5, 5, 6 (vii) 5, 5, 65, 5, 565 (viii) 9 75,,,, (ix), 5, 0 7 6,, (i) 65, 9 (ii) 5 9, (iii) 79 (iv) 0, 0, , (i),, 5, 7, 9; (ii),, 5, 07, ; (iii),, 6,, 0 ; ( ) (iv),,, 0, ; ( ) + 5, 9,, 7; 8 EXERCISE 9 5 (i) (ii) th (i) 0th (ii) 8th th st (i) 6r (ii) q b( r q) c( r p) 5 (i) m + p (ii) 0 6 p q (5 + 7) 8; 5 6 or 6 x = (i),, 8 (ii) ; 6 5 (i) 7 6 (ii) 5 56 (i) 5, 7, 9 (ii),, (iii) 7, 8, 9 (iv) 6, 0, (v), 6, 0, 57 (i) (i) 8080 (ii) ; Savig is importat for idividuals as it provides security agaist uexpected expeses It helps i the progress of the coutry ad the developmet of ecoomy 6 (i) 550 (ii) 75 (iii) 775; Keepig save eviromet ad coservatio of exhaustible resources i mid, the maually operated machie should be promoted so that eergy could be saved The maufacturer uses his/her wisdom to create more employmet for villagers by usig had operated machies
ARITHMETIC PROGRESSION
CHAPTER 5 ARITHMETIC PROGRESSION Poits to Remember :. A sequece is a arragemet of umbers or objects i a defiite order.. A sequece a, a, a 3,..., a,... is called a Arithmetic Progressio (A.P) if there exists
More informationSection 7 Fundamentals of Sequences and Series
ectio Fudametals of equeces ad eries. Defiitio ad examples of sequeces A sequece ca be thought of as a ifiite list of umbers. 0, , 0, , 0...,,,,,,. (iii),,,,... Defiitio: A sequece is a fuctio which
More information2.4  Sequences and Series
2.4  Sequeces ad Series Sequeces A sequece is a ordered list of elemets. Defiitio 1 A sequece is a fuctio from a subset of the set of itegers (usually either the set 80, 1, 2, 3,... < or the set 81, 2,
More informationA sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as
More informationSequences, Sums, and Products
CSCE 222 Discrete Structures for Computig Sequeces, Sums, ad Products Dr. Philip C. Ritchey Sequeces A sequece is a fuctio from a subset of the itegers to a set S. A discrete structure used to represet
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationMISCELLANEOUS SEQUENCES & SERIES QUESTIONS
MISCELLANEOUS SEQUENCES & SERIES QUESTIONS Questio (***+) Evaluate the followig sum 30 r ( 2) 4r 78. 3 MP2V, 75,822,200 Questio 2 (***+) Three umbers, A, B, C i that order, are i geometric progressio
More informationUNIT #5. Lesson #2 Arithmetic and Geometric Sequences. Lesson #3 Summation Notation. Lesson #4 Arithmetic Series. Lesson #5 Geometric Series
UNIT #5 SEQUENCES AND SERIES Lesso # Sequeces Lesso # Arithmetic ad Geometric Sequeces Lesso #3 Summatio Notatio Lesso #4 Arithmetic Series Lesso #5 Geometric Series Lesso #6 Mortgage Paymets COMMON CORE
More informationINFINITE SEQUENCES AND SERIES
11 INFINITE SEQUENCES AND SERIES INFINITE SEQUENCES AND SERIES 11.4 The Compariso Tests I this sectio, we will lear: How to fid the value of a series by comparig it with a kow series. COMPARISON TESTS
More informationRADICAL EXPRESSION. If a and x are real numbers and n is a positive integer, then x is an. n th root theorems: Example 1 Simplify
Example 1 Simplify 1.2A Radical Operatios a) 4 2 b) 16 1 2 c) 16 d) 2 e) 8 1 f) 8 What is the relatioship betwee a, b, c? What is the relatioship betwee d, e, f? If x = a, the x = = th root theorems: RADICAL
More informationSECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES
SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,
More information1. By using truth tables prove that, for all statements P and Q, the statement
Author: Satiago Salazar Problems I: Mathematical Statemets ad Proofs. By usig truth tables prove that, for all statemets P ad Q, the statemet P Q ad its cotrapositive ot Q (ot P) are equivalet. I example.2.3
More informationSolutions to Math 347 Practice Problems for the final
Solutios to Math 347 Practice Problems for the fial 1) True or False: a) There exist itegers x,y such that 50x + 76y = 6. True: the gcd of 50 ad 76 is, ad 6 is a multiple of. b) The ifiimum of a set is
More information11.1 Arithmetic Sequences and Series
11.1 Arithmetic Sequeces ad Series A itroductio 1, 4, 7, 10, 13 9, 1, 7, 15 6., 6.6, 7, 7.4 ππ+, 3, π+ 6 Arithmetic Sequeces ADD To get ext term 35 1 7. 3π + 9, 4, 8, 16, 3 9, 3, 1, 1/ 3 1,1/ 4,1/16,1/
More information(b) What is the probability that a particle reaches the upper boundary n before the lower boundary m?
MATH 529 The Boudary Problem The drukard s walk (or boudary problem) is oe of the most famous problems i the theory of radom walks. Oe versio of the problem is described as follows: Suppose a particle
More information62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +
62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of
More informationMatrices and vectors
Oe Matrices ad vectors This book takes for grated that readers have some previous kowledge of the calculus of real fuctios of oe real variable It would be helpful to also have some kowledge of liear algebra
More informationObjective Mathematics
. If sum of '' terms of a sequece is give by S Tr ( )( ), the 4 5 67 r (d) 4 9 r is equal to : T. Let a, b, c be distict ozero real umbers such that a, b, c are i harmoic progressio ad a, b, c are i arithmetic
More informationCALCULATION OF FIBONACCI VECTORS
CALCULATION OF FIBONACCI VECTORS Stuart D. Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithaca.edu ad Dai Novak Departmet of Mathematics, Ithaca College
More informationIn number theory we will generally be working with integers, though occasionally fractions and irrationals will come into play.
Number Theory Math 5840 otes. Sectio 1: Axioms. I umber theory we will geerally be workig with itegers, though occasioally fractios ad irratioals will come ito play. Notatio: Z deotes the set of all itegers
More informationMA131  Analysis 1. Workbook 2 Sequences I
MA3  Aalysis Workbook 2 Sequeces I Autum 203 Cotets 2 Sequeces I 2. Itroductio.............................. 2.2 Icreasig ad Decreasig Sequeces................ 2 2.3 Bouded Sequeces..........................
More informationSequences I. Chapter Introduction
Chapter 2 Sequeces I 2. Itroductio A sequece is a list of umbers i a defiite order so that we kow which umber is i the first place, which umber is i the secod place ad, for ay atural umber, we kow which
More informationThe picture in figure 1.1 helps us to see that the area represents the distance traveled. Figure 1: Area represents distance travelled
1 Lecture : Area Area ad distace traveled Approximatig area by rectagles Summatio The area uder a parabola 1.1 Area ad distace Suppose we have the followig iformatio about the velocity of a particle, how
More informationSeries III. Chapter Alternating Series
Chapter 9 Series III With the exceptio of the Null Sequece Test, all the tests for series covergece ad divergece that we have cosidered so far have dealt oly with series of oegative terms. Series with
More informationAPPENDIX F Complex Numbers
APPENDIX F Complex Numbers Operatios with Complex Numbers Complex Solutios of Quadratic Equatios Polar Form of a Complex Number Powers ad Roots of Complex Numbers Operatios with Complex Numbers Some equatios
More information2 n = n=1 a n is convergent and we let. i=1
Lecture 3 : Series So far our defiitio of a sum of umbers applies oly to addig a fiite set of umbers. We ca exted this to a defiitio of a sum of a ifiite set of umbers i much the same way as we exteded
More informationProof of Fermat s Last Theorem by Algebra Identities and Linear Algebra
Proof of Fermat s Last Theorem by Algebra Idetities ad Liear Algebra Javad Babaee Ragai Youg Researchers ad Elite Club, Qaemshahr Brach, Islamic Azad Uiversity, Qaemshahr, Ira Departmet of Civil Egieerig,
More informationComplex Analysis Spring 2001 Homework I Solution
Complex Aalysis Sprig 2001 Homework I Solutio 1. Coway, Chapter 1, sectio 3, problem 3. Describe the set of poits satisfyig the equatio z a z + a = 2c, where c > 0 ad a R. To begi, we see from the triagle
More informationSection 6.3: Geometric Sequences
40 Chpter 6 Sectio 6.: Geometric Sequeces My jobs offer ul costoflivig icrese to keep slries cosistet with ifltio. Suppose, for exmple, recet college grdute fids positio s sles mger erig ul slry of $6,000.
More informationMath 2112 Solutions Assignment 5
Math 2112 Solutios Assigmet 5 5.1.1 Idicate which of the followig relatioships are true ad which are false: a. Z Q b. R Q c. Q Z d. Z Z Z e. Q R Q f. Q Z Q g. Z R Z h. Z Q Z a. True. Every positive iteger
More informationMath 475, Problem Set #12: Answers
Math 475, Problem Set #12: Aswers A. Chapter 8, problem 12, parts (b) ad (d). (b) S # (, 2) = 2 2, sice, from amog the 2 ways of puttig elemets ito 2 distiguishable boxes, exactly 2 of them result i oe
More informationRecurrence Relations
Recurrece Relatios Aalysis of recursive algorithms, such as: it factorial (it ) { if (==0) retur ; else retur ( * factorial()); } Let t be the umber of multiplicatios eeded to calculate factorial(). The
More information4.1 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 informationComplex Numbers Solutions
Complex Numbers Solutios Joseph Zoller February 7, 06 Solutios. (009 AIME I Problem ) There is a complex umber with imagiary part 64 ad a positive iteger such that Fid. [Solutio: 697] 4i + + 4i. 4i 4i
More informationDirichlet s Theorem on Arithmetic Progressions
Dirichlet s Theorem o Arithmetic Progressios Athoy Várilly Harvard Uiversity, Cambridge, MA 0238 Itroductio Dirichlet s theorem o arithmetic progressios is a gem of umber theory. A great part of its beauty
More informationChapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
More informationa 3, a 4, ... are the terms of the sequence. The number a n is the nth term of the sequence, and the entire sequence is denoted by a n
60_090.qxd //0 : PM Page 59 59 CHAPTER 9 Ifiite Series Sectio 9. EXPLORATION Fidig Patters Describe a patter for each of the followig sequeces. The use your descriptio to write a formula for the th term
More informationNumber of fatalities X Sunday 4 Monday 6 Tuesday 2 Wednesday 0 Thursday 3 Friday 5 Saturday 8 Total 28. Day
LECTURE # 8 Mea Deviatio, Stadard Deviatio ad Variace & Coefficiet of variatio Mea Deviatio Stadard Deviatio ad Variace Coefficiet of variatio First, we will discuss it for the case of raw data, ad the
More informationSection 11.8: Power Series
Sectio 11.8: Power Series 1. Power Series I this sectio, we cosider geeralizig the cocept of a series. Recall that a series is a ifiite sum of umbers a. We ca talk about whether or ot it coverges ad i
More information10.1 Sequences. n term. We will deal a. a n or a n n. ( 1) n ( 1) n 1 2 ( 1) a =, 0 0,,,,, ln n. n an 2. n term.
0. Sequeces A sequece is a list of umbers writte i a defiite order: a, a,, a, a is called the first term, a is the secod term, ad i geeral eclusively with ifiite sequeces ad so each term Notatio: the sequece
More informationMIXED REVIEW of Problem Solving
MIXED REVIEW of Problem Solvig STATE TEST PRACTICE classzoe.com Lessos 2.4 2.. MULTISTEP PROBLEM A ball is dropped from a height of 2 feet. Each time the ball hits the groud, it bouces to 70% of its previous
More informationANOTHER GENERALIZED FIBONACCI SEQUENCE 1. INTRODUCTION
ANOTHER GENERALIZED FIBONACCI SEQUENCE MARCELLUS E. WADDILL A N D LOUIS SACKS Wake Forest College, Wisto Salem, N. C., ad Uiversity of ittsburgh, ittsburgh, a. 1. INTRODUCTION Recet issues of umerous periodicals
More informationEcon 325/327 Notes on Sample Mean, Sample Proportion, Central Limit Theorem, Chisquare Distribution, Student s t distribution 1.
Eco 325/327 Notes o Sample Mea, Sample Proportio, Cetral Limit Theorem, Chisquare Distributio, Studet s t distributio 1 Sample Mea By Hiro Kasahara We cosider a radom sample from a populatio. Defiitio
More informationTEACHER CERTIFICATION STUDY GUIDE
COMPETENCY 1. ALGEBRA SKILL 1.1 1.1a. ALGEBRAIC STRUCTURES Kow why the real ad complex umbers are each a field, ad that particular rigs are ot fields (e.g., itegers, polyomial rigs, matrix rigs) Algebra
More information1. Hilbert s Grand Hotel. The Hilbert s Grand Hotel has infinite many rooms numbered 1, 2, 3, 4
. Hilbert s Grad Hotel The Hilbert s Grad Hotel has ifiite may rooms umbered,,,.. Situatio. The Hotel is full ad a ew guest arrives. Ca the mager accommodate the ew guest?  Yes, he ca. There is a simple
More informationMa 530 Infinite Series I
Ma 50 Ifiite Series I Please ote that i additio to the material below this lecture icorporated material from the Visual Calculus web site. The material o sequeces is at Visual Sequeces. (To use this li
More informationOnce we have a sequence of numbers, the next thing to do is to sum them up. Given a sequence (a n ) n=1
. Ifiite Series Oce we have a sequece of umbers, the ext thig to do is to sum them up. Give a sequece a be a sequece: ca we give a sesible meaig to the followig expressio? a = a a a a While summig ifiitely
More informationAppendix F: Complex Numbers
Appedix F Complex Numbers F1 Appedix F: Complex Numbers Use the imagiary uit i to write complex umbers, ad to add, subtract, ad multiply complex umbers. Fid complex solutios of quadratic equatios. Write
More informationZeros of Polynomials
Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree
More informationADDITIONAL MATHEMATICS FORM 5 MODULE 2
PROGRAM DIDIK CEMERLANG AKADEMIK SPM ADDITIONAL MATHEMATICS FORM 5 MODULE 2 PROGRESSIONS (Geometric Progressio) ORGANISED BY: JABATAN PELAJARAN NEGERI PULAU PINANG CHAPTER 2 : GEOMETRIC PROGRESSIONS Cotets
More information3.2 Properties of Division 3.3 Zeros of Polynomials 3.4 Complex and Rational Zeros of Polynomials
Math 60 www.timetodare.com 3. Properties of Divisio 3.3 Zeros of Polyomials 3.4 Complex ad Ratioal Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered
More informationThe Ratio Test. THEOREM 9.17 Ratio Test Let a n be a series with nonzero terms. 1. a. n converges absolutely if lim. n 1
460_0906.qxd //04 :8 PM Page 69 SECTION 9.6 The Ratio ad Root Tests 69 Sectio 9.6 EXPLORATION Writig a Series Oe of the followig coditios guaratees that a series will diverge, two coditios guaratee that
More informationsin(n) + 2 cos(2n) n 3/2 3 sin(n) 2cos(2n) n 3/2 a n =
60. Ratio ad root tests 60.1. Absolutely coverget series. Defiitio 13. (Absolute covergece) A series a is called absolutely coverget if the series of absolute values a is coverget. The absolute covergece
More informationPart A, for both Section 200 and Section 501
Istructios Please write your solutios o your ow paper. These problems should be treated as essay questios. A problem that says give a example or determie requires a supportig explaatio. I all problems,
More information5 Sequences and Series
Bria E. Veitch 5 Sequeces ad Series 5. Sequeces A sequece is a list of umbers i a defiite order. a is the first term a 2 is the secod term a is the th term The sequece {a, a 2, a 3,..., a,..., } is a
More informationSOLVED EXAMPLES
Prelimiaries Chapter PELIMINAIES Cocept of Divisibility: A ozero iteger t is said to be a divisor of a iteger s if there is a iteger u such that s tu I this case we write t s (i) 6 as ca be writte as
More informationREVISION SHEET FP1 (MEI) ALGEBRA. Identities In mathematics, an identity is a statement which is true for all values of the variables it contains.
the Further Mathematics etwork wwwfmetworkorguk V 07 The mai ideas are: Idetities REVISION SHEET FP (MEI) ALGEBRA Before the exam you should kow: If a expressio is a idetity the it is true for all values
More informationBINOMIAL COEFFICIENT AND THE GAUSSIAN
BINOMIAL COEFFICIENT AND THE GAUSSIAN The biomial coefficiet is defied as! k!(! ad ca be writte out i the form of a Pascal Triagle startig at the zeroth row with elemet 0,0) ad followed by the two umbers,
More informationand the sum of its first n terms be denoted by. Convergence: An infinite series is said to be convergent if, a definite unique number., finite.
INFINITE SERIES Seqece: If a set of real mbers a seqece deoted by * + * Or * + * occr accordig to some defiite rle, the it is called + if is fiite + if is ifiite Series: is called a series ad is deoted
More informationHow to Maximize a Function without Really Trying
How to Maximize a Fuctio without Really Tryig MARK FLANAGAN School of Electrical, Electroic ad Commuicatios Egieerig Uiversity College Dubli We will prove a famous elemetary iequality called The Rearragemet
More informationQuiz No. 1. ln n n. 1. Define: an infinite sequence A function whose domain is N 2. Define: a convergent sequence A sequence that has a limit
Quiz No.. Defie: a ifiite sequece A fuctio whose domai is N 2. Defie: a coverget sequece A sequece that has a limit 3. Is this sequece coverget? Why or why ot? l Yes, it is coverget sice L=0 by LHR. INFINITE
More informationa for a 1 1 matrix. a b a b 2 2 matrix: We define det ad bc 3 3 matrix: We define a a a a a a a a a a a a a a a a a a
Math E2b Lecture #8 Notes This week is all about determiats. We ll discuss how to defie them, how to calculate them, lear the allimportat property kow as multiliearity, ad show that a square matrix A
More informationStudents will calculate quantities that involve positive and negative rational exponents.
: Ratioal Expoets What are ad? Studet Outcomes Studets will calculate quatities that ivolve positive ad egative ratioal expoets. Lesso Notes Studets exted their uderstadig of iteger expoets to ratioal
More informationLINEAR RECURSION RELATIONS  LESSON FOUR SECONDORDER LINEAR RECURSION RELATIONS
LINEAR RECURSION RELATIONS  LESSON FOUR SECONDORDER LINEAR RECURSION RELATIONS BROTHER ALFRED BROUSSEAU St. Mary's College, Califoria Give a secodorder liear recursio relatio (.1) T. 1 = a T + b T 1,
More informationw (1) ˆx w (1) x (1) /ρ and w (2) ˆx w (2) x (2) /ρ.
2 5. Weighted umber of late jobs 5.1. Release dates ad due dates: maximimizig the weight of otime jobs Oce we add release dates, miimizig the umber of late jobs becomes a sigificatly harder problem. For
More informationChapter 6 Principles of Data Reduction
Chapter 6 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 0 Chapter 6 Priciples of Data Reductio Sectio 6. Itroductio Goal: To summarize or reduce the data X, X,, X to get iformatio about a
More informationAlgebra II Notes Unit Seven: Powers, Roots, and Radicals
Syllabus Objectives: 7. The studets will use properties of ratioal epoets to simplify ad evaluate epressios. 7.8 The studet will solve equatios cotaiig radicals or ratioal epoets. b a, the b is the radical.
More informationThe Random Walk For Dummies
The Radom Walk For Dummies Richard A Mote Abstract We look at the priciples goverig the oedimesioal discrete radom walk First we review five basic cocepts of probability theory The we cosider the Beroulli
More informationIs mathematics discovered or
996 Chapter 1 Sequeces, Iductio, ad Probability Sectio 1. Objectives Evaluate a biomial coefficiet. Expad a biomial raised to a power. Fid a particular term i a biomial expasio. The Biomial Theorem Galaxies
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationElementary Statistics
Elemetary Statistics M. Ghamsary, Ph.D. Sprig 004 Chap 0 Descriptive Statistics Raw Data: Whe data are collected i origial form, they are called raw data. The followig are the scores o the first test of
More informationSequences of Definite Integrals, Factorials and Double Factorials
47 6 Joural of Iteger Sequeces, Vol. 8 (5), Article 5.4.6 Sequeces of Defiite Itegrals, Factorials ad Double Factorials Thierry DaaPicard Departmet of Applied Mathematics Jerusalem College of Techology
More informationPUTNAM TRAINING INEQUALITIES
PUTNAM TRAINING INEQUALITIES (Last updated: December, 207) Remark This is a list of exercises o iequalities Miguel A Lerma Exercises If a, b, c > 0, prove that (a 2 b + b 2 c + c 2 a)(ab 2 + bc 2 + ca
More informationP1 Chapter 8 :: Binomial Expansion
P Chapter 8 :: Biomial Expasio jfrost@tiffi.kigsto.sch.uk www.drfrostmaths.com @DrFrostMaths Last modified: 6 th August 7 Use of DrFrostMaths for practice Register for free at: www.drfrostmaths.com/homework
More informationCommutativity in Permutation Groups
Commutativity i Permutatio Groups Richard Wito, PhD Abstract I the group Sym(S) of permutatios o a oempty set S, fixed poits ad trasiet poits are defied Prelimiary results o fixed ad trasiet poits are
More informationMA131  Analysis 1. Workbook 9 Series III
MA3  Aalysis Workbook 9 Series III Autum 004 Cotets 4.4 Series with Positive ad Negative Terms.............. 4.5 Alteratig Series.......................... 4.6 Geeral Series.............................
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.436J/15.085J Fall 2008 Lecture 19 11/17/2008 LAWS OF LARGE NUMBERS II THE STRONG LAW OF LARGE NUMBERS
MASSACHUSTTS INSTITUT OF TCHNOLOGY 6.436J/5.085J Fall 2008 Lecture 9 /7/2008 LAWS OF LARG NUMBRS II Cotets. The strog law of large umbers 2. The Cheroff boud TH STRONG LAW OF LARG NUMBRS While the weak
More informationCIS Spring 2018 (instructor Val Tannen)
CIS 160  Sprig 2018 (istructor Val Tae) Lecture 5 Thursday, Jauary 25 COUNTING We cotiue studyig how to use combiatios ad what are their properties. Example 5.1 How may 8letter strigs ca be costructed
More informationSolutions to home assignments (sketches)
Matematiska Istitutioe Peter Kumli 26th May 2004 TMA401 Fuctioal Aalysis MAN670 Applied Fuctioal Aalysis 4th quarter 2003/2004 All documet cocerig the course ca be foud o the course home page: http://www.math.chalmers.se/math/grudutb/cth/tma401/
More informationLecture 6 Chi Square Distribution (χ 2 ) and Least Squares Fitting
Lecture 6 Chi Square Distributio (χ ) ad Least Squares Fittig Chi Square Distributio (χ ) Suppose: We have a set of measuremets {x 1, x, x }. We kow the true value of each x i (x t1, x t, x t ). We would
More informationProbability theory and mathematical statistics:
N.I. Lobachevsky State Uiversity of Nizhi Novgorod Probability theory ad mathematical statistics: Law of Total Probability. Associate Professor A.V. Zorie Law of Total Probability. 1 / 14 Theorem Let H
More informationCALCULATING FIBONACCI VECTORS
THE GENERALIZED BINET FORMULA FOR CALCULATING FIBONACCI VECTORS Stuart D Aderso Departmet of Physics, Ithaca College 953 Daby Road, Ithaca NY 14850, USA email: saderso@ithacaedu ad Dai Novak Departmet
More informationProperties and Hypothesis Testing
Chapter 3 Properties ad Hypothesis Testig 3.1 Types of data The regressio techiques developed i previous chapters ca be applied to three differet kids of data. 1. Crosssectioal data. 2. Time series data.
More informationNotes on iteration and Newton s method. Iteration
Notes o iteratio ad Newto s method Iteratio Iteratio meas doig somethig over ad over. I our cotet, a iteratio is a sequece of umbers, vectors, fuctios, etc. geerated by a iteratio rule of the type 1 f
More informationMath 140A Elementary Analysis Homework Questions 1
Math 14A Elemetary Aalysis Homewor Questios 1 1 Itroductio 1.1 The Set N of Natural Numbers 1 Prove that 1 2 2 2 2 1 ( 1(2 1 for all atural umbers. 2 Prove that 3 11 (8 5 4 2 for all N. 4 (a Guess a formula
More informationTHIS paper analyzes the behavior of those complex
IAENG Iteratioal Joural of Computer Sciece 39:4 IJCS_39_4_6 Itrisic Order Lexicographic Order Vector Order ad Hammig Weight Luis Gozález Abstract To compare biary tuple probabilities with o eed to compute
More informationA Proof of Birkhoff s Ergodic Theorem
A Proof of Birkhoff s Ergodic Theorem Joseph Hora September 2, 205 Itroductio I Fall 203, I was learig the basics of ergodic theory, ad I came across this theorem. Oe of my supervisors, Athoy Quas, showed
More informationDISTRIBUTION LAW Okunev I.V.
1 DISTRIBUTION LAW Okuev I.V. Distributio law belogs to a umber of the most complicated theoretical laws of mathematics. But it is also a very importat practical law. Nothig ca help uderstad complicated
More informationEssential Question How can you use properties of exponents to simplify products and quotients of radicals?
. TEXAS ESSENTIAL KNOWLEDGE AND SKILLS A.7.G Properties of Ratioal Expoets ad Radicals Essetial Questio How ca you use properties of expoets to simplify products ad quotiets of radicals? Reviewig Properties
More informationCS / MCS 401 Homework 3 grader solutions
CS / MCS 401 Homework 3 grader solutios assigmet due July 6, 016 writte by Jāis Lazovskis maximum poits: 33 Some questios from CLRS. Questios marked with a asterisk were ot graded. 1 Use the defiitio of
More informationRiemann 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, oegative fuctio o the closed iterval [a, b] Fid
More informationPhysics 324, Fall Dirac Notation. These notes were produced by David Kaplan for Phys. 324 in Autumn 2001.
Physics 324, Fall 2002 Dirac Notatio These otes were produced by David Kapla for Phys. 324 i Autum 2001. 1 Vectors 1.1 Ier product Recall from liear algebra: we ca represet a vector V as a colum vector;
More informationBasic Sets. Functions. MTH299  Examples. Example 1. Let S = {1, {2, 3}, 4}. Indicate whether each statement is true or false. (a) S = 4. (e) 2 S.
Basic Sets Example 1. Let S = {1, {2, 3}, 4}. Idicate whether each statemet is true or false. (a) S = 4 (b) {1} S (c) {2, 3} S (d) {1, 4} S (e) 2 S. (f) S = {1, 4, {2, 3}} (g) S Example 2. Compute the
More informationT1.1 Lesson 3  Arithmetic & Geometric Series & Summation Notation
Fast Five T. Lesso 3  Arithmetic & Geometric eries & ummatio Notatio Math L  atowski Fid the sum of the first 00 umbers Outlie a way to solve this problem ad the carry out your pla Fid the sum of the
More informationAddition: Property Name Property Description Examples. a+b = b+a. a+(b+c) = (a+b)+c
Notes for March 31 Fields: A field is a set of umbers with two (biary) operatios (usually called additio [+] ad multiplicatio [ ]) such that the followig properties hold: Additio: Name Descriptio Commutativity
More informationAnalysis of Experimental Data
Aalysis of Experimetal Data 6544597.0479 ± 0.000005 g Quatitative Ucertaity Accuracy vs. Precisio Whe we make a measuremet i the laboratory, we eed to kow how good it is. We wat our measuremets to be both
More informationMatsubaraGreen s Functions
MatsubaraGree s Fuctios Time Orderig : Cosider the followig operator If H = H the we ca trivially factorise this as, E(s = e s(h+ E(s = e sh e s I geeral this is ot true. However for practical applicatio
More informationNew Results for the Fibonacci Sequence Using Binet s Formula
Iteratioal Mathematical Forum, Vol. 3, 208, o. 6, 29266 HIKARI Ltd, www.mhikari.com https://doi.org/0.2988/imf.208.832 New Results for the Fiboacci Sequece Usig Biet s Formula Reza Farhadia Departmet
More informationECE 901 Lecture 4: Estimation of Lipschitz smooth functions
ECE 9 Lecture 4: Estiatio of Lipschitz sooth fuctios R. Nowak 5/7/29 Cosider the followig settig. Let Y f (X) + W, where X is a rado variable (r.v.) o X [, ], W is a r.v. o Y R, idepedet of X ad satisfyig
More information2.4.2 A Theorem About Absolutely Convergent Series
0 Versio of August 27, 200 CHAPTER 2. INFINITE SERIES Add these two series: + 3 2 + 5 + 7 4 + 9 + 6 +... = 3 l 2. (2.20) 2 Sice the reciprocal of each iteger occurs exactly oce i the last series, we would
More information