Induction. Induction and Recursion. Induction is a very useful proof technique

By the end of this section you will be able to prove the Chinese Remainder Theorem apply this theorem to solve simultaneous linear congruences

Chapte : Theoy of Modula Aithmetic 8 Sectio D Chiese Remaide Theoem By the ed of this sectio you will be able to pove the Chiese Remaide Theoem apply this theoem to solve simultaeous liea cogueces The

Algoritmalar, Tamsayılar ve Matrisler

Algoitml, Tmsyıl ve Mtisle CSC-59 Ayık Ypıl Kostti Busch - LSU f Foksiyolı Büyümesi : R R g : R R f ( is O( g( Big-Oh: E kötü duum Big-Omeg: f ( is ( g( E iyi duum Big-Thet: f ( is ( g( Ayı duum Kostti

Introduction Common Divisors. Discrete Mathematics Andrei Bulatov

Intoduction Common Divisos Discete Mathematics Andei Bulatov Discete Mathematics Common Divisos 3- Pevious Lectue Integes Division, popeties of divisibility The division algoithm Repesentation of numbes

Recursion. Algorithm : Design & Analysis [3]

Recusio Algoithm : Desig & Aalysis [] I the last class Asymptotic gowth ate he Sets Ο, Ω ad Θ Complexity Class A Example: Maximum Susequece Sum Impovemet of Algoithm Compaiso of Asymptotic Behavio Aothe

BINOMIAL THEOREM SOLUTION. 1. (D) n. = (C 0 + C 1 x +C 2 x C n x n ) (1+ x+ x 2 +.)

BINOMIAL THEOREM SOLUTION. (D) ( + + +... + ) (+ + +.) The coefficiet of + + + +... + fo. Moeove coefficiet of is + + + +... + if >. So. (B)... e!!!! The equied coefficiet coefficiet of i e -.!...!. (A),

PROGRESSION AND SERIES

INTRODUCTION PROGRESSION AND SERIES A gemet of umbes {,,,,, } ccodig to some well defied ule o set of ules is clled sequece Moe pecisely, we my defie sequece s fuctio whose domi is some subset of set of

MATH 118 HW 7 KELLY DOUGAN, ANDREW KOMAR, MARIA SIMBIRSKY, BRANDEN LASKE

MATH 118 HW 7 KELLY DOUGAN, ANDREW KOMAR, MARIA SIMBIRSKY, BRANDEN LASKE Prt 1. Let be odd rime d let Z such tht gcd(, 1. Show tht if is qudrtic residue mod, the is qudrtic residue mod for y ositive iteger.

Counting Functions and Subsets

CHAPTER 1 Coutig Fuctios ad Subsets This chapte of the otes is based o Chapte 12 of PJE See PJE p144 Hee ad below, the efeeces to the PJEccles book ae give as PJE The goal of this shot chapte is to itoduce

2002 Quarter 1 Math 172 Final Exam. Review

00 Qute Mth 7 Fil Exm. Review Sectio.. Sets Repesettio of Sets:. Listig the elemets. Set-uilde Nottio Checkig fo Memeship (, ) Compiso of Sets: Equlity (=, ), Susets (, ) Uio ( ) d Itesectio ( ) of Sets

Fermat s Little Theorem. mod 13 = 0, = }{{} mod 13 = 0. = a a a }{{} mod 13 = a 12 mod 13 = 1, mod 13 = a 13 mod 13 = a.

Departmet of Mathematical Scieces Istructor: Daiva Puciskaite Discrete Mathematics Fermat s Little Theorem 43.. For all a Z 3, calculate a 2 ad a 3. Case a = 0. 0 0 2-times Case a 0. 0 0 3-times a a 2-times

ICS141: Discrete Mathematics for Computer Science I

ICS4: Discrete Mthemtics for Computer Sciece I Dept. Iformtio & Computer Sci., J Stelovsky sed o slides y Dr. Bek d Dr. Still Origils y Dr. M. P. Frk d Dr. J.L. Gross Provided y McGrw-Hill 3- Quiz. gcd(84,96).

EXERCISE - 01 CHECK YOUR GRASP

EXERISE - 0 HEK YOUR GRASP 3. ( + Fo sum of coefficiets put ( + 4 ( + Fo sum of coefficiets put ; ( + ( 4. Give epessio c e ewitte s 7 4 7 7 3 7 7 ( 4 3( 4... 7( 4 7 7 7 3 ( 4... 7( 4 Lst tem ecomes (4

Chapter 7. Kleene s Theorem. 7.1 Kleene s Theorem. The following theorem is the most important and fundamental result in the theory of FA s:

Chpte 7 Kleene s Theoem 7.1 Kleene s Theoem The following theoem is the most impotnt nd fundmentl esult in the theoy of FA s: Theoem 6 Any lnguge tht cn e defined y eithe egul expession, o finite utomt,

M3P14 EXAMPLE SHEET 1 SOLUTIONS

M3P14 EXAMPLE SHEET 1 SOLUTIONS 1. Show tht for, b, d itegers, we hve (d, db) = d(, b). Sice (, b) divides both d b, d(, b) divides both d d db, d hece divides (d, db). O the other hd, there exist m d

5. Solving recurrences

5. Solvig recurreces Time Complexity Alysis of Merge Sort T( ) 0 if 1 2T ( / 2) otherwise sortig oth hlves mergig Q. How to prove tht the ru-time of merge sort is O( )? A. 2 Time Complexity Alysis of Merge

Chapter 3: Theory of Modular Arithmetic 38

Chapte 3: Theoy of Modula Aithmetic 38 Section D Chinese Remainde Theoem By the end of this section you will be able to pove the Chinese Remainde Theoem apply this theoem to solve simultaneous linea conguences

For this purpose, we need the following result:

9 Lectue Sigulities of omplex Fuctio A poit is clled sigulity of fuctio f ( z ) if f ( z ) is ot lytic t the poit. A sigulity is clled isolted sigulity of f ( z ), if f ( z ) is lytic i some puctued disk

Similar idea to multiplication in N, C. Divide and conquer approach provides unexpected improvements. Naïve matrix multiplication

Next. Covered bsics of simple desig techique (Divided-coquer) Ch. of the text.. Next, Strsse s lgorithm. Lter: more desig d coquer lgorithms: MergeSort. Solvig recurreces d the Mster Theorem. Similr ide

Semiconductors materials

Semicoductos mteils Elemetl: Goup IV, Si, Ge Biy compouds: III-V (GAs,GSb, ISb, IP,...) IV-VI (PbS, PbSe, PbTe,...) II-VI (CdSe, CdTe,...) Tey d Qutey compouds: G x Al -x As, G x Al -x As y P -y III IV

Using Difference Equations to Generalize Results for Periodic Nested Radicals

Usig Diffeece Equatios to Geealize Results fo Peiodic Nested Radicals Chis Lyd Uivesity of Rhode Islad, Depatmet of Mathematics South Kigsto, Rhode Islad 2 2 2 2 2 2 2 π = + + +... Vieta (593) 2 2 2 =

SOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES

#A17 INTEGERS 9 2009), 181-190 SOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES Deick M. Keiste Depatmet of Mathematics, Pe State Uivesity, Uivesity Pak, PA 16802 dmk5075@psu.edu James A. Selles Depatmet

M5. LTI Systems Described by Linear Constant Coefficient Difference Equations

5. LTI Systes Descied y Lie Costt Coefficiet Diffeece Equtios Redig teil: p.34-4, 245-253 3/22/2 I. Discete-Tie Sigls d Systes Up til ow we itoduced the Fouie d -tsfos d thei popeties with oly ief peview

Lower Bounds for Cover-Free Families

Loe Bouds fo Cove-Fee Families Ali Z. Abdi Covet of Nazaeth High School Gade, Abas 7, Haifa Nade H. Bshouty Dept. of Compute Sciece Techio, Haifa, 3000 Apil, 05 Abstact Let F be a set of blocks of a t-set

We show that every analytic function can be expanded into a power series, called the Taylor series of the function.

10 Lectue 8 We show tht evey lytic fuctio c be expded ito powe seies, clled the Tylo seies of the fuctio. Tylo s Theoem: Let f be lytic i domi D & D. The, f(z) c be expessed s the powe seies f( z) b (

ANSWER KEY PHYSICS. Workdone X

ANSWER KEY PHYSICS 6 6 6 7 7 7 9 9 9 0 0 0 CHEMISTRY 6 6 6 7 7 7 9 9 9 0 0 60 MATHEMATICS 6 66 7 76 6 6 67 7 77 7 6 6 7 7 6 69 7 79 9 6 70 7 0 90 PHYSICS F L l. l A Y l A ;( A R L L A. W = (/ lod etesio

Chapter 7 Infinite Series

MA Ifiite Series Asst.Prof.Dr.Supree Liswdi Chpter 7 Ifiite Series Sectio 7. Sequece A sequece c be thought of s list of umbers writte i defiite order:,,...,,... 2 The umber is clled the first term, 2

Divisibility. c = bf = (ae)f = a(ef) EXAMPLE: Since 7 56 and , the Theorem above tells us that

Divisibility DEFINITION: If a and b ae integes with a 0, we say that a divides b if thee is an intege c such that b = ac. If a divides b, we also say that a is a diviso o facto of b. NOTATION: d n means

Exam 2 CMSC 203 Fall 2009 Name SOLUTION KEY Show All Work! 1. (16 points) Circle T if the corresponding statement is True or F if it is False.

1 (1 poits) Circle T if the correspodig statemet is True or F if it is False T F For ay positive iteger,, GCD(, 1) = 1 T F Every positive iteger is either prime or composite T F If a b mod p, the (a/p)

Lacunary Weak I-Statistical Convergence

Ge. Mat. Notes, Vol. 8, No., May 05, pp. 50-58 ISSN 9-784; Copyigt ICSRS Publicatio, 05 www.i-css.og vailable ee olie at ttp//www.gema.i Lacuay Wea I-Statistical Covegece Haize Gümüş Faculty o Eegli Educatio,

Lecture 6: October 16, 2017

Ifomatio ad Codig Theoy Autum 207 Lectue: Madhu Tulsiai Lectue 6: Octobe 6, 207 The Method of Types Fo this lectue, we will take U to be a fiite uivese U, ad use x (x, x 2,..., x to deote a sequece of

Crosscorrelation of m-sequences, Exponential sums and Dickson

Cosscoelatio o m-equeces, Epoetial sums ad Dicso polyomials To Helleseth Uiesity o Bege NORWAY Joit wo with Aia Johase ad Aleade Kholosha Itoductio Outlie m-sequeces Coelatio o sequeces Popeties o m-sequeces

On the k-lucas Numbers of Arithmetic Indexes

Alied Mthetics 0 3 0-06 htt://d.doi.og/0.436/.0.307 Published Olie Octobe 0 (htt://www.scirp.og/oul/) O the -ucs Nubes of Aithetic Idees Segio lco Detet of Mthetics d Istitute fo Alied Micoelectoics (IUMA)

Merge Sort. Outline and Reading. Divide-and-Conquer. Divide-and-conquer paradigm ( 4.1.1) Merge-sort ( 4.1.1)

Merge Sort 7 2 9 4 2 4 7 9 7 2 2 7 9 4 4 9 7 7 2 2 9 9 4 4 Merge Sort versio 1.3 1 Outlie d Redig Divide-d-coquer prdigm ( 4.1.1 Merge-sort ( 4.1.1 Algorithm Mergig two sorted sequeces Merge-sort tree

Mathematical Statistics

7 75 Ode Sttistics The ode sttistics e the items o the dom smple ed o odeed i mitude om the smllest to the lest Recetl the impotce o ode sttistics hs icesed owi to the moe equet use o opmetic ieeces d

We will begin by supplying the proof to (a).

(The solutios of problem re mostly from Jeffrey Mudrock s HWs) Problem 1. There re three sttemet from Exmple 5.4 i the textbook for which we will supply proofs. The sttemets re the followig: () The spce

PLANCESS RANK ACCELERATOR

PLANCESS RANK ACCELERATOR MATHEMATICS FOR JEE MAIN & ADVANCED Sequeces d Seies 000questios with topic wise execises 000 polems of IIT-JEE & AIEEE exms of lst yes Levels of Execises ctegoized ito JEE Mi

Interpolation. 1. What is interpolation?

Iterpoltio. Wht is iterpoltio? A uctio is ote give ol t discrete poits such s:.... How does oe id the vlue o t other vlue o? Well cotiuous uctio m e used to represet the + dt vlues with pssig through the

Technical Report: Bessel Filter Analysis

Sasa Mahmoodi 1 Techical Repot: Bessel Filte Aalysis 1 School of Electoics ad Compute Sciece, Buildig 1, Southampto Uivesity, Southampto, S17 1BJ, UK, Email: sm3@ecs.soto.ac.uk I this techical epot, we

«A first lesson on Mathematical Induction»

Bcgou ifotio: «A fist lesso o Mtheticl Iuctio» Mtheticl iuctio is topic i H level Mthetics It is useful i Mtheticl copetitios t ll levels It hs bee coo sight tht stuets c out the poof b theticl iuctio,

Induction: Solutions

Writig Proofs Misha Lavrov Iductio: Solutios Wester PA ARML Practice March 6, 206. Prove that a 2 2 chessboard with ay oe square removed ca always be covered by shaped tiles. Solutio : We iduct o. For

Mathematical Induction (selected questions)

Mtheticl Iductio (selected questios). () Let P() e the propositio : For P(), L.H.S. R.H.S., P() is true. Assue P() is true for soe turl uer, tht is, () For P( ),, y () By the Priciple of Mtheticl Iductio,

Induction proofs - practice! SOLUTIONS

Iductio proofs - practice! SOLUTIONS 1. Prove that f ) = 6 + + 15 is odd for all Z +. Base case: For = 1, f 1) = 41) + 1) + 13 = 19. Sice 19 is odd, f 1) is odd - base case prove. Iductive hypothesis:

CSE 1400 Applied Discrete Mathematics Number Theory and Proofs

CSE 1400 Applied Discrete Mathematics Number Theory ad Proofs Departmet of Computer Scieces College of Egieerig Florida Tech Sprig 01 Problems for Number Theory Backgroud Number theory is the brach of

Trial division, Pollard s p 1, Pollard s ρ, and Fermat s method. Christopher Koch 1. April 8, 2014

Iteger Divisio Algorithm ad Cogruece Iteger Trial divisio,,, ad with itegers mod Iverses mod Multiplicatio ad GCD Iteger Christopher Koch 1 1 Departmet of Computer Sciece ad Egieerig CSE489/589 Algorithms

Numbers (Part I) -- Solutions

Ley College -- For AMATYC SML Mth Competitio Cochig Sessios v.., [/7/00] sme s /6/009 versio, with presettio improvemets Numbers Prt I) -- Solutios. The equtio b c 008 hs solutio i which, b, c re distict

INFINITE SERIES. ,... having infinite number of terms is called infinite sequence and its indicated sum, i.e., a 1

Appedix A.. Itroductio As discussed i the Chpter 9 o Sequeces d Series, sequece,,...,,... hvig ifiite umber of terms is clled ifiite sequece d its idicted sum, i.e., + + +... + +... is clled ifite series

Graphing Review Part 3: Polynomials

Grphig Review Prt : Polomils Prbols Recll, tht the grph of f ( ) is prbol. It is eve fuctio, hece it is smmetric bout the bout the -is. This mes tht f ( ) f ( ). Its grph is show below. The poit ( 0,0)

CHAPTER 5 : SERIES. 5.2 The Sum of a Series Sum of Power of n Positive Integers Sum of Series of Partial Fraction Difference Method

CHAPTER 5 : SERIES 5.1 Seies 5. The Sum of a Seies 5..1 Sum of Powe of Positive Iteges 5.. Sum of Seies of Patial Factio 5..3 Diffeece Method 5.3 Test of covegece 5.3.1 Divegece Test 5.3. Itegal Test 5.3.3

Relating to, connected or concerned with, quality or qualities. Now usually in implied or expressed opposition to quantitative.

. Mathematical bacgou I you chose poessio, it will be ecessay to mae egieeig esig ecisios. Whe it comes to pogammig, you will ote have a selectio o possible algoithms o ata stuctues; howeve, whe you compae

= 5! 3! 2! = 5! 3! (5 3)!. In general, the number of different groups of r items out of n items (when the order is ignored) is given by n!

0 Combiatoial Aalysis Copyight by Deiz Kalı 4 Combiatios Questio 4 What is the diffeece betwee the followig questio i How may 3-lette wods ca you wite usig the lettes A, B, C, D, E ii How may 3-elemet

Conditional Convergence of Infinite Products

Coditioal Covegece of Ifiite Poducts William F. Tech Ameica Mathematical Mothly 106 1999), 646-651 I this aticle we evisit the classical subject of ifiite poducts. Fo stadad defiitios ad theoems o this

a) The average (mean) of the two fractions is halfway between them: b) The answer is yes. Assume without loss of generality that p < r.

Solutios to MAML Olympiad Level 00. Factioated a) The aveage (mea) of the two factios is halfway betwee them: p ps+ q ps+ q + q s qs qs b) The aswe is yes. Assume without loss of geeality that p

SUTCLIFFE S NOTES: CALCULUS 2 SWOKOWSKI S CHAPTER 11

UTCLIFFE NOTE: CALCULU WOKOWKI CHAPTER Ifiite eries Coverget or Diverget eries Cosider the sequece If we form the ifiite sum 0, 00, 000, 0 00 000, we hve wht is clled ifiite series We wt to fid the sum

[ 20 ] 1. Inequality exists only between two real numbers (not complex numbers). 2. If a be any real number then one and only one of there hold.

[ 0 ]. Iequlity eists oly betwee two rel umbers (ot comple umbers).. If be y rel umber the oe d oly oe of there hold.. If, b 0 the b 0, b 0.. (i) b if b 0 (ii) (iii) (iv) b if b b if either b or b b if

On the Basis Property of Eigenfunction. of the Frankl Problem with Nonlocal Parity Conditions. of the Third Kind

It.. Cotemp. Math. Scieces Vol. 9 o. 3 33-38 HIKARI Lt www.m-hikai.com http://x.oi.og/.988/ijcms..33 O the Basis Popety o Eigeuctio o the Fakl Poblem with Nolocal Paity Coitios o the Thi Ki A. Sameipou

MATH Midterm Solutions

MATH 2113 - Midtem Solutios Febuay 18 1. A bag of mables cotais 4 which ae ed, 4 which ae blue ad 4 which ae gee. a How may mables must be chose fom the bag to guaatee that thee ae the same colou? We ca

Pre-Calculus - Chapter 3 Sections Notes

Pre-Clculus - Chpter 3 Sectios 3.1-3.4- Notes Properties o Epoets (Review) 1. ( )( ) = + 2. ( ) =, (c) = 3. 0 = 1 4. - = 1/( ) 5. 6. c Epoetil Fuctios (Sectio 3.1) Deiitio o Epoetil Fuctios The uctio deied

Disjoint Sets { 9} { 1} { 11} Disjoint Sets (cont) Operations. Disjoint Sets (cont) Disjoint Sets (cont) n elements

Disjoit Sets elemets { x, x, } X =, K Opeatios x Patitioed ito k sets (disjoit sets S, S,, K Fid-Set(x - etu set cotaiig x Uio(x,y - make a ew set by combiig the sets cotaiig x ad y (destoyig them S k

Summary: Binomial Expansion...! r. where

Summy: Biomil Epsio 009 M Teo www.techmejcmth-sg.wes.com ) Re-cp of Additiol Mthemtics Biomil Theoem... whee )!!(! () The fomul is ville i MF so studets do ot eed to memoise it. () The fomul pplies oly

Convergence rates of approximate sums of Riemann integrals

Covergece rtes of pproximte sums of Riem itegrls Hiroyuki Tski Grdute School of Pure d Applied Sciece, Uiversity of Tsuku Tsuku Irki 5-857 Jp tski@mth.tsuku.c.jp Keywords : covergece rte; Riem sum; Riem

Data Structures. Element Uniqueness Problem. Hash Tables. Example. Hash Tables. Dana Shapira. 19 x 1. ) h(x 4. ) h(x 2. ) h(x 3. h(x 1. x 4. x 2.

Element Uniqueness Poblem Dt Stuctues Let x,..., xn < m Detemine whethe thee exist i j such tht x i =x j Sot Algoithm Bucket Sot Dn Shpi Hsh Tbles fo (i=;i

Math 4318 : Real Analysis II Mid-Term Exam 1 14 February 2013

Mth 4318 : Rel Anlysis II Mid-Tem Exm 1 14 Febuy 2013 Nme: Definitions: Tue/Flse: Poofs: 1. 2. 3. 4. 5. 6. Totl: Definitions nd Sttements of Theoems 1. (2 points) Fo function f(x) defined on (, b) nd fo

Some Properties of the K-Jacobsthal Lucas Sequence

Deepia Jhala et. al. /Iteatioal Joual of Mode Scieces ad Egieeig Techology (IJMSET) ISSN 349-3755; Available at https://www.imset.com Volume Issue 3 04 pp.87-9; Some Popeties of the K-Jacobsthal Lucas

Discrete Mathematics I Tutorial 12

Discrete Mthemtics I Tutoril Refer to Chpter 4., 4., 4.4. For ech of these sequeces fid recurrece reltio stisfied by this sequece. (The swers re ot uique becuse there re ifiitely my differet recurrece

UNIT V: Z-TRANSFORMS AND DIFFERENCE EQUATIONS. Dr. V. Valliammal Department of Applied Mathematics Sri Venkateswara College of Engineering

UNIT V: -TRANSFORMS AND DIFFERENCE EQUATIONS D. V. Vllimml Deptmet of Applied Mthemtics Si Vektesw College of Egieeig TOPICS:. -Tsfoms Elemet popeties.. Ivese -Tsfom usig ptil fctios d esidues. Covolutio

9.1 The multiplicative group of a finite field. Theorem 9.1. The multiplicative group F of a finite field is cyclic.

Chapte 9 Pimitive Roots 9.1 The multiplicative goup of a finite fld Theoem 9.1. The multiplicative goup F of a finite fld is cyclic. Remak: In paticula, if p is a pime then (Z/p) is cyclic. In fact, this

The Area of a Triangle

The e of Tingle tkhlid June 1, 015 1 Intodution In this tile we will e disussing the vious methods used fo detemining the e of tingle. Let [X] denote the e of X. Using se nd Height To stt off, the simplest

Mathematical Induction. How do you program a robot to climb infinite stairs? Let s use that as a proof method. 22c:19 Chapter 4 Hantao Zhang

Mathematical Iductio c:9 Chapter 4 Hatao Zhag How do you program a robot to climb ifiite stairs? First, you let the robot get to the base platform of the staircase The repeat: From the curret positio,

Geometric Sequences. Geometric Sequence. Geometric sequences have a common ratio.

s A geometric sequece is sequece such tht ech successive term is obtied from the previous term by multiplyig by fixed umber clled commo rtio. Exmples, 6, 8,, 6,..., 0, 0, 0, 80,... Geometric sequeces hve

These Two Weeks. Mathematical Induction Chapter 4 Lecture 10 & Lecture 13. CPRE 310 Discrete Mathematics. Mathematical Induction Problems

These Two Wees CPRE 0 Discrete Mathematics Mathematical Iductio Chapter Lecture 0 & Lecture Lecture 0 Mathematical Iductio (MI) MI Problems Lecture, Review Lecture February 9, Test I A list of topics ad

MATH 304: MIDTERM EXAM SOLUTIONS

MATH 304: MIDTERM EXAM SOLUTIONS [The problems are each worth five poits, except for problem 8, which is worth 8 poits. Thus there are 43 possible poits.] 1. Use the Euclidea algorithm to fid the greatest

CH 39 USING THE GCF TO REDUCE FRACTIONS

359 CH 39 USING THE GCF TO EDUCE FACTIONS educig Algeric Frctios M ost of us lered to reduce rithmetic frctio dividig the top d the ottom of the frctio the sme (o-zero) umer. For exmple, 30 30 5 75 75

Recurrece reltio & Recursio 9 Chpter 3 Recurrece reltio & Recursio Recurrece reltio : A recurrece is equtio or iequlity tht describes fuctio i term of itself with its smller iputs. T T The problem of size

A note on random minimum length spanning trees

A ote o adom miimum legth spaig tees Ala Fieze Miklós Ruszikó Lubos Thoma Depatmet of Mathematical Scieces Caegie Mello Uivesity Pittsbugh PA15213, USA ala@adom.math.cmu.edu, usziko@luta.sztaki.hu, thoma@qwes.math.cmu.edu

2012 GCE A Level H2 Maths Solution Paper Let x,

GCE A Level H Maths Solutio Pape. Let, y ad z be the cost of a ticet fo ude yeas, betwee ad 5 yeas, ad ove 5 yeas categoies espectively. 9 + y + 4z =. 7 + 5y + z = 8. + 4y + 5z = 58.5 Fo ude, ticet costs

When two numbers are written as the product of their prime factors, they are in factored form.

10 1 Study Guide Pages 420 425 Factos Because 3 4 12, we say that 3 and 4 ae factos of 12. In othe wods, factos ae the numbes you multiply to get a poduct. Since 2 6 12, 2 and 6 ae also factos of 12. The

Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,

MSS SEQUENCE AND SERIES CA SEQUENCE A sequece is fuctio of tul ubes with codoi is the set of el ubes (Coplex ubes. If Rge is subset of el ubes (Coplex ubes the it is clled el sequece (Coplex sequece. Exple

Finite q-identities related to well-known theorems of Euler and Gauss. Johann Cigler

Fiite -idetities elated to well-ow theoems of Eule ad Gauss Joha Cigle Faultät fü Mathemati Uivesität Wie A-9 Wie, Nodbegstaße 5 email: oha.cigle@uivie.ac.at Abstact We give geealizatios of a fiite vesio

The Pigeonhole Principle 3.4 Binomial Coefficients

Discete M athematic Chapte 3: Coutig 3. The Pigeohole Piciple 3.4 Biomial Coefficiets D Patic Cha School of Compute Sciece ad Egieeig South Chia Uivesity of Techology Ageda Ch 3. The Pigeohole Piciple

A proof of the binomial theorem

A poof of the binomial theoem If n is a natual numbe, let n! denote the poduct of the numbes,2,3,,n. So! =, 2! = 2 = 2, 3! = 2 3 = 6, 4! = 2 3 4 = 24 and so on. We also let 0! =. If n is a non-negative

Homework 1 Solutions CSE 101 Summer 2017

Homewok 1 Soutions CSE 101 Summe 2017 1 Waming Up 1.1 Pobem and Pobem Instance Find the smaest numbe in an aay of n integes a 1, a 2,..., a n. What is the input? What is the output? Is this a pobem o a

Sect Simplifying Radical Expressions. We can use our properties of exponents to establish two properties of radicals: and

128 Sect 10.3 - Simplifyig Rdicl Expressios Cocept #1 Multiplictio d Divisio Properties of Rdicls We c use our properties of expoets to estlish two properties of rdicls: () 1/ 1/ 1/ & ( Multiplictio d

MATH /19: problems for supervision in week 08 SOLUTIONS

MATH10101 2018/19: poblems fo supevisio i week 08 Q1. Let A be a set. SOLUTIONS (i Pove that the fuctio c: P(A P(A, defied by c(x A \ X, is bijective. (ii Let ow A be fiite, A. Use (i to show that fo each

Week 13 Notes: 1) Riemann Sum. Aim: Compute Area Under a Graph. Suppose we want to find out the area of a graph, like the one on the right:

Week 1 Notes: 1) Riem Sum Aim: Compute Are Uder Grph Suppose we wt to fid out the re of grph, like the oe o the right: We wt to kow the re of the red re. Here re some wys to pproximte the re: We cut the

Solutions 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

Eigenfunction Expansion. For a given function on the internal a x b the eigenfunction expansion of f(x):

Eigefuctio Epsio: For give fuctio o the iterl the eigefuctio epsio of f(): f ( ) cmm( ) m 1 Eigefuctio Epsio (Geerlized Fourier Series) To determie c s we multiply oth sides y Φ ()r() d itegrte: f ( )

ON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS

Joual of Pue ad Alied Mathematics: Advaces ad Alicatios Volume 0 Numbe 03 Pages 5-58 ON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS ALI H HAKAMI Deatmet of Mathematics

(II.G) PRIME POWER MODULI AND POWER RESIDUES

II.G PRIME POWER MODULI AND POWER RESIDUES I II.C, we used the Chiese Remider Theorem to reduce cogrueces modulo m r i i to cogrueces modulo r i i. For exmles d roblems, we stuck with r i 1 becuse we hd

Einstein Classes, Unit No. 102, 103, Vardhman Ring Road Plaza, Vikas Puri Extn., Outer Ring Road New Delhi , Ph. : ,

MB BINOMIAL THEOREM Biomial Epessio : A algebaic epessio which cotais two dissimila tems is called biomial epessio Fo eample :,,, etc / ( ) Statemet of Biomial theoem : If, R ad N, the : ( + ) = a b +

Lecture 4 Recursive Algorithm Analysis. Merge Sort Solving Recurrences The Master Theorem

Lecture 4 Recursive Algorithm Alysis Merge Sort Solvig Recurreces The Mster Theorem Merge Sort MergeSortA, left, right) { if left < right) { mid = floorleft + right) / 2); MergeSortA, left, mid); MergeSortA,

0 otherwise. sin( nx)sin( kx) 0 otherwise. cos( nx) sin( kx) dx 0 for all integers n, k.

. Computtio of Fourier Series I this sectio, we compute the Fourier coefficiets, f ( x) cos( x) b si( x) d b, i the Fourier series To do this, we eed the followig result o the orthogolity of the trigoometric

Expansion by Laguerre Function for Wave Diffraction around an Infinite Cylinder

Joul of Applied Mthemtics d Physics, 5, 3, 75-8 Published Olie Juy 5 i SciRes. http://www.scip.og/joul/jmp http://dx.doi.og/.436/jmp.5.3 Expsio by Lguee Fuctio fo Wve Diffctio oud Ifiite Cylide Migdog

( ) ( ) ( ) ( ) Solved Examples. JEE Main/Boards = The total number of terms in the expansion are 8.

Mathematics. Solved Eamples JEE Mai/Boads Eample : Fid the coefficiet of y i c y y Sol: By usig fomula of fidig geeal tem we ca easily get coefficiet of y. I the biomial epasio, ( ) th tem is c T ( y )

Week 8. Topic 2 Properties of Logarithms

Week 8 Topic 2 Popeties of Logithms 1 Week 8 Topic 2 Popeties of Logithms Intoduction Since the esult of ithm is n eponent, we hve mny popeties of ithms tht e elted to the popeties of eponents. They e

PROGRESSIONS AND SERIES

PROGRESSIONS AND SERIES A sequece is lso clled progressio. We ow study three importt types of sequeces: () The Arithmetic Progressio, () The Geometric Progressio, () The Hrmoic Progressio. Arithmetic Progressio.

x a y n + b = 1 0<b a, n > 0 (1.1) x 1 - a y = b 0<b a, n > 0 (1.1') b n sin 2 + cos 2 = 1 x n = = cos 2 6 Superellipse (Lamé curve)

6 Supeellipse (Lmé cuve) 6. Equtios of supeellipse A supeellipse (hoizotlly log) is epessed s follows. Implicit Equtio y + b 0 0 (.) Eplicit Equtio y b - 0 0 (.') Whe 3, b, the supeellipses fo

Chapter Real Numbers

Chpter. - Rel Numbers Itegers: coutig umbers, zero, d the egtive of the coutig umbers. ex: {,-3, -, -,,,, 3, } Rtiol Numbers: quotiets of two itegers with ozero deomitor; termitig or repetig decimls. ex:

MTH 146 Class 16 Notes

MTH 46 Clss 6 Notes 0.4- Cotiued Motivtio: We ow cosider the rc legth of polr curve. Suppose we wish to fid the legth of polr curve curve i terms of prmetric equtios s: r f where b. We c view the cos si

Convergence rates of approximate sums of Riemann integrals

Jourl of Approximtio Theory 6 (9 477 49 www.elsevier.com/locte/jt Covergece rtes of pproximte sums of Riem itegrls Hiroyuki Tski Grdute School of Pure d Applied Sciece, Uiversity of Tsukub, Tsukub Ibrki

GRAPHING LINEAR EQUATIONS. Linear Equations. x l ( 3,1 ) _x-axis. Origin ( 0, 0 ) Slope = change in y change in x. Equation for l 1.

GRAPHING LINEAR EQUATIONS Qudrt II Qudrt I ORDERED PAIR: The first umer i the ordered pir is the -coordite d the secod umer i the ordered pir is the y-coordite. (, ) Origi ( 0, 0 ) _-is Lier Equtios Qudrt