BULLETIN OF MATHEMATICS AND STATISTICS RESEARCH

Size: px
Start display at page:

Download "BULLETIN OF MATHEMATICS AND STATISTICS RESEARCH"

Transcription

1 Vol..Issue.1.14 BULLETIN OF MATHEMATICS AND STATISTICS RESEARCH A eer Reviewed Itertiol Reserch Jourl RESEARCH ARTICLE INTEGRAL SOLUTIONS OF TERNARY QUADRATIC DIOHANTINE EQUATIONS 7X Y 135Z S.RIYA 1 M.A.GOALAN S.VIDHYALAKSHMI 3 1 M.hil studet Deprtmet of MthemticsShrimti Idir Gdhi College Trich Tmildu Idi 3 rofessor Deprtmet of Mthemtics Shrimti Idir Gdhi CollegeTrich Tmildu Idi ABSTRACT The terr qudrtic homogeeous equtio represetig coe give b 7X Y 135Z is led for its o-ero distict iteger poits o it. Si differet ptters of iteger solutios stisfig the coe uder cosidertio re give. A few iterestig reltios betwee the solutios d specil umber ptters re preseted. Give itegrl solutio o the cosidered coe three triples of itegers geerted from the give solutio re ehibited. * S.RIYA Author for Correspodece Article Ifo: Article received :8/1/13 Revised o:15//14 Accepted o:18//14 Kewords: Terr qudrtic itegrl solutios 1 Mthemtics Subject Clssifictio:11D9 Nottios: m : rmid umber of rk with sie m T m : olgol umber of rk with sie m INTRODUCTON The terr qudrtic Diophtie equtios offer ulimited field for reserch due to their vriet [ 1]. For etesive review of vrious problems. Oe m refer [19]. This commuictio cocers with 7X Y 135Z et other iterestig terr qudrtic equtio represetig coe for determiig its ifiitel m o-ero itegrl poits. Also few iterestig reltios mog the solutios d specil umbers re preseted. METHOD OF ANALYSIS: The terr qudrtic equtio to be solved to be give b 3

2 S.RIYA et l (1) It is see tht (1) is stisfied b ( ) ( ) d ( ) However we hve other choices of solutios which re preseted below. Itroducig the lier trsformtios X T X 7T () i (1) it is writte s.1 choice: 1 Let 14t 15 (3) 14b b Write 15 s 15 (1 i 14)(1 i (4) 14) Substitutig (4) d (5) i (3) d emploig the method of fctoritio defie ( i 14t) (1 i 14)( i 14b) Equtig rel d imgir prts we get X X( b) 14b 8b (6) T T( b) 14b b Usig (6) i () we hve 3 4b 4b 6 84b 4b (7) Thus (7) d (4) represet the o-ero distict itegrl solutios of (1). roperties: ( 1) T 8 ( Mod) ( 1) ( 1) T 1 1 b) T 3 1( Mod13) ( b 7(mod1) 5 ( ( 1)) 168T 3 48p. Choice: Equtio (3) c be writte s 15 14t is Nst umber (8) Tke 14 t 15 (9) Substitutig (9) i (8) it becomes 1 (1) Which is stisfied b pq 1p q 1p q (11) Thus from (11) (9) d () represets the correspodig itegrl solutios of (1) re 63p q 6pq 16p 8q 1pq (1) 1p q 8pq (5) Vol..Issue

3 S.RIYA et l. roperties: ( p1) T p T p ( p1) T p T6 ( p) T 64 p 4 p 4(mod56) 1(mod36) 1(mod688) T 14 q (1 q) 165( Mod19).3 Choice:3 Equtio () is rewritte i the form of rtio s X Z Z T 14( Z T) Q X Z Q which is equivlet to the followig two equtios. QX Z( Q ) T X Z( 14Q ) 14QT Emploig the method of cross multiplictio we get X 14Q 8Q (14) T 14Q Q Z 14Q (15) (13) Usig the vlues of X d T i () we hve 3 4Q 4Q (16) 6 84Q 4Q Thus (16) (15) represet the o-ero itegrl solutios of (1). roperties: ( p1) T p ( p1) T 1 8 p 14(mod35) 16(mod6) ( p p 1) ( p p 1) 18T 4 p 6{ ( p p 1)) 8T 3 p is Nst umber. Note: Equtio () is lso writte i the form of rtio i three more ws s below X Z Z T 1. Q 14( Z T) X Z Q X Z 7( Z T). Q ( Z T) X Z Q X Z ( Z T) 3. Q 7( Z T) X Z Q (19) Followig the procedure preseted bove i ptter 3 re m set 3 more differet choices of iteger solutios to (1) Cosiderig (16) the correspodig iteger solutios of (1) re 4p 3q 4pq 84p 6q 4pq (17) (18) Vol..Issue

4 S.RIYA et l. 1. roperties: 14p q ( 1) ( ( p1) T p T46 ( 1 q) T q 4 p 4 ( p p) T 3 p 1) 18p (mod) (mod14) 5 3(mod64) Cosiderig (17) the correspodig iteger solutios of (1) re 6p 1q 4pq 1p 4q 4pq p 7q. roperties: ( ( 1) ) ( ( 1) ) 54p ( p) T ( q) T 14 p 86 q 31(mod53) 48(mod15) 3 5 ( p p 1)) 84T 3 p 48p 6p is Nst umber. Cosiderig (18) the correspodig iteger solutios of (1) re 1p 6q 4pq 1p 1q 4pq 3. roperties: 7p q ( p1) T p T4 ( p p) T p p (mod8) 3{ ( p 1) p) 8 T 3 p { ( p p 1)) 48T 1(mod5) is Nst umber. 84p 5 3 p is Nst umber. 3. REMARKABLE OBSERVATION: Let ) be the positive iitil solutio of (1).The ech of the followig three triples of ( itegers bsed o lso stisf (1). Triple:1 i which 1 ( ) 4 7 ( ) 1 15 Triple : ( ) 1 ( ) Vol..Issue

5 S.RIYA et l. 7 1 ( ) i which Triple : 3 7 ( ) ( ( ) ) {(4 14 ) ( 4( )) 18 1 {( 14( )) (14 4 ) Y 18 9 i which CONCLUSION: I this disserttio the terr qudrtic Diophtie equtios refferig coe is lsed for is o-ero distict itegrl oits. A few iterestig properties betwee the solutios d specil umbers re preseted. To coclude oe m serch for other ptters of solutios d their correspodig properties for the coe uder cosidertio REFERENCES [1]. Dickso L.E. Histor of Theor of Numbers Vol.Chelse ublishig comp NewYork 195 []. Gopl M.A.dichevi V. Itegrl solutio of terr qudrtic equtio ( ) 4 Actocieci Idic 8 Vol. XXXIVM No [3]. Gopl M.A.Klig Ri J. Observtio o the Diophtie equtio J.sci tech ; 8Vol () D Impct [4]. Gopl M.A.dichevi V. o terr qudrtic equtio 1 Impct J.sci tech; 8 Vol () [5]. Gopl M.A. Mju somth VithN. Itegrl solutios of terr qudrtic Diophtie equtio ( k 1). Impct J.sci tech; 8 Vol (4) [6]. Gopl M.A. Mju somth Itegrl solutio of terr qudrtic Diophtie equtio AtrticJMth 81-55(1). [7]. GoplM.A.d GmA. thgore trigles d specil polgol umbers Itertiol Jourl of Mthemticl Sciece J-Ju 1Vol.(9)No [8]. Gopl M.A. d Vijskr A.Observtios o thgore problem Act Cieci Idic 1Vol.XXXVIM No [9]. Gopl.M.A. d pdichelvi.v. Itegrl solutios of terr qudrtic equtio ( ) 4 Impct J.sci TSech; 11 Vol (5)No [1]. Gopl M.A.Klig Ri J.O terr qudrtic equtio 8 Impct J.sci tech ; 11 Vol (5) o [11]. Gopl M.A. Geeth D. Lttice poits o the hperbolid of two sheets Impct J.sci tech ; 1 Vol(4)No Vol..Issue

6 S.RIYA et l. [1]. Gopl M.A. Vidhlkshmi S. d Kvith A. Itegrl poits o the homogeeous Coe 7 DiophtusJ.Mth. 11() [13]. Gopl M.A. Vidhlkshmi S. SumthiG. Lttice poits o the hperboloid oe sheet DiophtusJ.mth. 11() [14]. Gopl M.A. Vidhlkshmi S. d LkshmiK. Itegrl poits o the hperboloid of two sheets DiophtusJ.mth. 11() [15]. Gopl M.A. d SrividhG. Observtios o [16]. GoplM.A. SgeethG.Observtio o 36. Archimedes J.Mth 1 (1) 3 AtrcticJ.Mth 19(4) 359- [17]. GoplM.A. d VijlkshmiR. O the terr qudrtic equtio ( 1)( ) >1 Bessel J.Mth 1() [18]. Mjusomth SgeethG. GoplM.A. O the homogeeous terr qudrtic Diophtie equtio ( k 1) ( k 1) Bessel J.Mth 1() [19]. Mjusomth SgeethG. GoplM.A. Observtios o the terr qudrtic equtio 3 Bessel J.Mth 1() []. Mordell L.J. Diophtie equtios Acdemic press New York 1969 Vol..Issue

INTEGRAL SOLUTIONS OF THE TERNARY CUBIC EQUATION

INTEGRAL SOLUTIONS OF THE TERNARY CUBIC EQUATION Itertiol Reserch Jourl of Egieerig d Techology IRJET) e-issn: 9-006 Volume: 04 Issue: Mr -017 www.irjet.et p-issn: 9-007 INTEGRL SOLUTIONS OF THE TERNRY CUBIC EQUTION y ) 4y y ) 97z G.Jki 1, C.Sry,* ssistt

More information

On The Homogeneous Quintic Equation with Five Unknowns

On The Homogeneous Quintic Equation with Five Unknowns IOSR Jourl of Mthemtics (IOSR-JM) e-issn: 78-78,p-ISSN: 319-76X, Volume 7, Issue 3 (Jul. - Aug. 013), PP 7-76 www.iosrjourls.org O The Homogeeous Quitic Equtio with Five Ukows y y 3 3 ( y ) 3(( y)( z w

More information

OBSERVATIONS ON TERNARY QUADRATIC DIOPHANTINE EQUATION - x 63y

OBSERVATIONS ON TERNARY QUADRATIC DIOPHANTINE EQUATION - x 63y OBSERVATIONS ON TERNARY QUADRATIC DIOPHANTINE EQUATION - x 63 K.Lakshmi R.Someshwari Asst.Professor, Departmet of Maematics, Shrimati Idira Gadhi College,Tamil adu,idia, M.Phil Scholar, Departmet of Maematics,

More information

Observations on the Non-homogeneous Quintic Equation with Four Unknowns

Observations on the Non-homogeneous Quintic Equation with Four Unknowns Itertiol Jourl of Mthemtic Reerch. ISSN 976-84 Volume, Number 1 (13), pp. 17-133 Itertiol Reerch Publictio Houe http://www.irphoue.com Obervtio o the No-homogeeou Quitic Equtio with Four Ukow S. Vidhylkhmi

More information

OBSERVATIONS ON THE NON- HOMOGENEOUS SEXTIC EQUATION WITH FOUR UNKNOWNS

OBSERVATIONS ON THE NON- HOMOGENEOUS SEXTIC EQUATION WITH FOUR UNKNOWNS ISSN: 1-87 Itertiol Jourl of Iovtive Reerch i Sciece, Egieerig d Techology Vol., Iue, My 01 OSERVTIONS ON THE NON- HOMOGENEOUS SEXTIC EQUTION WITH FOUR UNKNOWNS y ( k z S.Vidhylkhmi 1,M..Gopl,.Kvith Profeor,

More information

degree non-homogeneous Diophantine equation in six unknowns represented by x y 2z

degree non-homogeneous Diophantine equation in six unknowns represented by x y 2z Scholrs Jourl of Egieerig d Techology (SJET Sch. J. Eg. Tech., ; (A:97- Scholrs Acdeic d Scietific Pulisher (A Itertiol Pulisher for Acdeic d Scietific Resources www.sspulisher.co ISSN -X (Olie ISSN 7-9

More information

- Pyramidal number of rank n with size m. - Polygonal number of rank n with size m.

- Pyramidal number of rank n with size m. - Polygonal number of rank n with size m. Scholars Joural of Egieerig ad Techology SJET Sch. J. Eg. Tech., 0; :9-3 Scholars cademic ad Scietific Publisher Iteratioal Publisher for cademic ad Scietific Resources www.saspublisher.com ISSN 3-35X

More information

International journal of Engineering Research-Online A Peer Reviewed International Journal Articles available online

International journal of Engineering Research-Online A Peer Reviewed International Journal Articles available online Ieraioal joural of Egieerig Research-Olie A Peer Reviewed Ieraioal Joural Aricles available olie hp://www.ijoer.i Vol.., Issue.., 3 RESEARCH ARTICLE INTEGRAL SOLUTION OF 3 G.AKILA, M.A.GOPALAN, S.VIDHYALAKSHMI

More information

On Homogeneous Ternary Quadratic Diophantine Equation

On Homogeneous Ternary Quadratic Diophantine Equation IOSR Joural of Matheatics (IOSR-JM) e-issn:78-578, p-issn: 9-765X.Volue,Issue Ver. III (Ja.-Feb.6)PP 78-8 www.iosrjourals.org O Hoogeeous Terar Quadratic Diophatie Equatio 5 x R.buselvi ad N.hila Departet

More information

Research Article. ISSN (Print) *Corresponding author C.Nithya Abstract: The binary quadratic equation x 5xy

Research Article. ISSN (Print) *Corresponding author C.Nithya   Abstract: The binary quadratic equation x 5xy Scholars Joural o Egieerig ad Techology (SJET) Sch. J. Eg. Tech., 04; (B):6-30 Scholars Academic ad Scietiic Publisher (A Iteratioal Publisher or Academic ad Scietiic Resources) www.saspublisher.com ISSN

More information

ONTERNARY QUADRATIC DIOPHANTINE EQUATION 2x 2 + 3y 2 = 4z

ONTERNARY QUADRATIC DIOPHANTINE EQUATION 2x 2 + 3y 2 = 4z ONTERNARY QUADRATIC DIOPHANTINE EQUATION x + 3y = 4z M.A.Gopala 1, K.Geetha ad Maju Somaath 3 1 Professor,Dept.of Mathematics,Shrimati Idira Gadhi College,Trichy- 0 Tamiladu, Asst Professor,Dept.of Mathematics,Cauvery

More information

ON THE BINARY QUADRATIC DIOPHANTINE EQUATION

ON THE BINARY QUADRATIC DIOPHANTINE EQUATION Iteratioal Joural of Scietific Egieerig ad Applied Sciece (IJSEAS) - Volume-, Issue-4, Jue 0 ISSN: 9-470 ON THE BINARY QUADRATIC DIOPHANTINE EQUATION - 6 0 M.A.Gopala * V.Geetha, D.Priaka. Professor, Departmet

More information

A GENERALIZATION OF GAUSS THEOREM ON QUADRATIC FORMS

A GENERALIZATION OF GAUSS THEOREM ON QUADRATIC FORMS A GENERALIZATION OF GAU THEOREM ON QUADRATIC FORM Nicole I Brtu d Adi N Cret Deprtmet of Mth - Criov Uiversity, Romi ABTRACT A origil result cocerig the extesio of Guss s theorem from the theory of biry

More information

International Research Journal of Engineering and Technology (IRJET) e-issn: ON THE BINARY QUADRATIC DIOPHANTINE EQUATION

International Research Journal of Engineering and Technology (IRJET) e-issn: ON THE BINARY QUADRATIC DIOPHANTINE EQUATION Iteratioal Research Joural of Egieerig ad Techolog (IRJET) e-issn: 9-006 Volume: 0 Issue: 04 Jul-0 www.irjet.et p-issn: 9-007 ON THE BINARY QUADRATIC DIOPHANTINE EQUATION - + 8 = 0 S.Vidhalakshmi, M.A.Gopala,

More information

International Journal of Multidisciplinary Research and Development. M.A. Gopalan, A. Kavitha, G. Thamaraiselvi

International Journal of Multidisciplinary Research and Development. M.A. Gopalan, A. Kavitha, G. Thamaraiselvi Volume:, Issue: 6, -7 Jue 015.allsubjectjoural.com e-issn: 49-418 p-issn: 49-5979 Impact Factor:.76 M.A. Gopala Professor, Departmet of Idira Gadhi College, Trichy-6000, Tamiladu, Idia. A. Kavitha Lecturer,

More information

Homogeneous Bi-Quadratic Equation With Five Unknowns

Homogeneous Bi-Quadratic Equation With Five Unknowns Interntionl Journl of Mthemtics Reserch. ISSN 0976-580 Volume 6, Number (01), pp. 5-51 Interntionl Reserch Publiction House http://www.irphouse.com Homogeneous Bi-Qudrtic Eqution With Five y p q Unknowns

More information

[ 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.

[ 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

More information

A GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD

A GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD Diol Bgoo () A GENERAL METHOD FOR SOLVING ORDINARY DIFFERENTIAL EQUATIONS: THE FROBENIUS (OR SERIES) METHOD I. Itroductio The first seprtio of vribles (see pplictios to Newto s equtios) is ver useful method

More information

Some New Iterative Methods Based on Composite Trapezoidal Rule for Solving Nonlinear Equations

Some New Iterative Methods Based on Composite Trapezoidal Rule for Solving Nonlinear Equations Itertiol Jourl of Mthemtics d Sttistics Ivetio (IJMSI) E-ISSN: 31 767 P-ISSN: 31-759 Volume Issue 8 August. 01 PP-01-06 Some New Itertive Methods Bsed o Composite Trpezoidl Rule for Solvig Nolier Equtios

More information

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

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

More information

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

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

More information

GLOBAL JOURNAL OF ENGINEERING SCIENCE AND RESEARCHES

GLOBAL JOURNAL OF ENGINEERING SCIENCE AND RESEARCHES GLOBAL JOURNAL OF ENGINEERING SCIENCE AND RESEARCHES A STUDY ON THE HYPERBOLA x 9 0 S. Vidhyalakshmi, M.A. Gopala & T. Mahalakshmi*, Professor, Departmet of Mathematics, Shrimati Idira Gadhi College, Trichy-60

More information

ON THE TERNARY QUADRATIC DIOPHANTINE EQUATION

ON THE TERNARY QUADRATIC DIOPHANTINE EQUATION Interntionl Reserch Journl of Engineering nd echnology IRJE) e-issn: 395-56 Volume: Issue: 4 July-15 www.irjet.net p-issn: 395-7 ON HE ERNARY QUADRAIC DIOPHANINE EQUAION 4x y ) 3xy 16 M.A.Gopln 1 *, S.Vidhylkshmi,

More information

Abstract We obtain infinitely many non-zero integer sextuples ( x, y, z, w, p, T )

Abstract We obtain infinitely many non-zero integer sextuples ( x, y, z, w, p, T ) IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SCIENCES & RESEARCH TECHNOLOGY O the No-Homogeeous Equatio of the Eighth Degree with Six Ukows x 5 -y 5 +(x 3 -y 3 xy = p(z -w T 3 S.Vidhyalakshmi *1, K.Lakshmi,

More information

Indices and Logarithms

Indices and Logarithms the Further Mthemtics etwork www.fmetwork.org.uk V 7 SUMMARY SHEET AS Core Idices d Logrithms The mi ides re AQA Ed MEI OCR Surds C C C C Lws of idices C C C C Zero, egtive d frctiol idices C C C C Bsic

More information

APPLICATION OF DIFFERENCE EQUATIONS TO CERTAIN TRIDIAGONAL MATRICES

APPLICATION OF DIFFERENCE EQUATIONS TO CERTAIN TRIDIAGONAL MATRICES Scietific Reserch of the Istitute of Mthetics d Coputer Sciece 3() 0, 5-0 APPLICATION OF DIFFERENCE EQUATIONS TO CERTAIN TRIDIAGONAL MATRICES Jolt Borows, Le Łcińs, Jowit Rychlews Istitute of Mthetics,

More information

SOLUTION OF DIFFERENTIAL EQUATION FOR THE EULER-BERNOULLI BEAM

SOLUTION OF DIFFERENTIAL EQUATION FOR THE EULER-BERNOULLI BEAM Jourl of Applied Mthemtics d Computtiol Mechics () 57-6 SOUION O DIERENIA EQUAION OR HE EUER-ERNOUI EAM Izbel Zmorsk Istitute of Mthemtics Czestochow Uiversit of echolog Częstochow Pold izbel.zmorsk@im.pcz.pl

More information

ALGEBRA. Set of Equations. have no solution 1 b1. Dependent system has infinitely many solutions

ALGEBRA. Set of Equations. have no solution 1 b1. Dependent system has infinitely many solutions Qudrtic Equtios ALGEBRA Remider theorem: If f() is divided b( ), the remider is f(). Fctor theorem: If ( ) is fctor of f(), the f() = 0. Ivolutio d Evlutio ( + b) = + b + b ( b) = + b b ( + b) 3 = 3 +

More information

1. (25 points) Use the limit definition of the definite integral and the sum formulas to compute. [1 x + x2

1. (25 points) Use the limit definition of the definite integral and the sum formulas to compute. [1 x + x2 Mth 3, Clculus II Fil Exm Solutios. (5 poits) Use the limit defiitio of the defiite itegrl d the sum formuls to compute 3 x + x. Check your swer by usig the Fudmetl Theorem of Clculus. Solutio: The limit

More information

The Elementary Arithmetic Operators of Continued Fraction

The Elementary Arithmetic Operators of Continued Fraction Americ-Eursi Jourl of Scietific Reserch 0 (5: 5-63, 05 ISSN 88-6785 IDOSI Pulictios, 05 DOI: 0.589/idosi.ejsr.05.0.5.697 The Elemetry Arithmetic Opertors of Cotiued Frctio S. Mugssi d F. Mistiri Deprtmet

More information

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

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

More information

Numbers (Part I) -- Solutions

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

More information

OBSERVATIONS ON THE NON- HOMOGENEOUS QUINTIC EQUATION WITH FIVE UNKNOWNS

OBSERVATIONS ON THE NON- HOMOGENEOUS QUINTIC EQUATION WITH FIVE UNKNOWNS OBSERVATIONS ON THE NON- HOMOGENEOUS QUINTIC EQUATION WITH FIVE UNKNOWNS y z w P S.Vidhylhmi S.Mlli M.A.Gopl Aitt Profeor of Mthemtic SIGC Trichy-TmildIdi Lectrer of Mthemtic SIGC Trichy-TmildIdi Aitt

More information

Section 6.3: Geometric Sequences

Section 6.3: Geometric Sequences 40 Chpter 6 Sectio 6.: Geometric Sequeces My jobs offer ul cost-of-livig 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 information

M3P14 EXAMPLE SHEET 1 SOLUTIONS

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

More information

Linear Programming. Preliminaries

Linear Programming. Preliminaries Lier Progrmmig Prelimiries Optimiztio ethods: 3L Objectives To itroduce lier progrmmig problems (LPP To discuss the stdrd d coicl form of LPP To discuss elemetry opertio for lier set of equtios Optimiztio

More information

Double Sums of Binomial Coefficients

Double Sums of Binomial Coefficients Itertiol Mthemticl Forum, 3, 008, o. 3, 50-5 Double Sums of Biomil Coefficiets Athoy Sofo School of Computer Sciece d Mthemtics Victori Uiversity, PO Box 448 Melboure, VIC 800, Austrli thoy.sofo@vu.edu.u

More information

Laws of Integral Indices

Laws of Integral Indices A Lws of Itegrl Idices A. Positive Itegrl Idices I, is clled the se, is clled the idex lso clled the expoet. mes times.... Exmple Simplify 5 6 c Solutio 8 5 6 c 6 Exmple Simplify Solutio The results i

More information

Objective Mathematics

Objective Mathematics . o o o o {cos 4 cos 9 si cos 65 } si 7º () cos 6º si 8º. If x R oe of these, the mximum vlue of the expressio si x si x.cos x c cos x ( c) is : () c c c c c c. If ( cos )cos cos ; 0, the vlue of 4. The

More information

BITSAT MATHEMATICS PAPER. If log 0.0( ) log 0.( ) the elogs to the itervl (, ] () (, ] [,+ ). The poit of itersectio of the lie joiig the poits i j k d i+ j+ k with the ple through the poits i+ j k, i

More information

Power Series Solutions to Generalized Abel Integral Equations

Power Series Solutions to Generalized Abel Integral Equations Itertiol Jourl of Mthemtics d Computtiol Sciece Vol., No. 5, 5, pp. 5-54 http://www.isciece.org/jourl/ijmcs Power Series Solutios to Geerlized Abel Itegrl Equtios Rufi Abdulli * Deprtmet of Physics d Mthemtics,

More information

Discrete Mathematics I Tutorial 12

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

More information

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

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.

More information

In an algebraic expression of the form (1), like terms are terms with the same power of the variables (in this case

In an algebraic expression of the form (1), like terms are terms with the same power of the variables (in this case Chpter : Algebr: A. Bckgroud lgebr: A. Like ters: I lgebric expressio of the for: () x b y c z x y o z d x... p x.. we cosider x, y, z to be vribles d, b, c, d,,, o,.. to be costts. I lgebric expressio

More information

Convergence rates of approximate sums of Riemann integrals

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

More information

Vectors. Vectors in Plane ( 2

Vectors. Vectors in Plane ( 2 Vectors Vectors i Ple ( ) The ide bout vector is to represet directiol force Tht mes tht every vector should hve two compoets directio (directiol slope) d mgitude (the legth) I the ple we preset vector

More information

Schrödinger Equation Via Laplace-Beltrami Operator

Schrödinger Equation Via Laplace-Beltrami Operator IOSR Jourl of Mthemtics (IOSR-JM) e-issn: 78-578, p-issn: 39-765X. Volume 3, Issue 6 Ver. III (Nov. - Dec. 7), PP 9-95 www.iosrjourls.org Schrödiger Equtio Vi Lplce-Beltrmi Opertor Esi İ Eskitşçioğlu,

More information

On A Subclass of Harmonic Univalent Functions Defined By Generalized Derivative Operator

On A Subclass of Harmonic Univalent Functions Defined By Generalized Derivative Operator Itertiol Jourl of Moder Egieerig Reserch (IJMER) Vol., Issue.3, My-Jue 0-56-569 ISSN: 49-6645 N. D. Sgle Dertmet of Mthemtics, Asheb Dge College of Egieerig, Asht, Sgli, (M.S) Idi 4630. Y. P. Ydv Dertmet

More information

EXERCISE a a a 5. + a 15 NEETIIT.COM

EXERCISE a a a 5. + a 15 NEETIIT.COM - Dowlod our droid App. Sigle choice Type Questios EXERCISE -. The first term of A.P. of cosecutive iteger is p +. The sum of (p + ) terms of this series c be expressed s () (p + ) () (p + ) (p + ) ()

More information

Lesson-2 PROGRESSIONS AND SERIES

Lesson-2 PROGRESSIONS AND SERIES Lesso- PROGRESSIONS AND SERIES Arithmetic Progressio: A sequece of terms is sid to be i rithmetic progressio (A.P) whe the differece betwee y term d its preceedig term is fixed costt. This costt is clled

More information

Students must always use correct mathematical notation, not calculator notation. the set of positive integers and zero, {0,1, 2, 3,...

Students must always use correct mathematical notation, not calculator notation. the set of positive integers and zero, {0,1, 2, 3,... Appedices Of the vrious ottios i use, the IB hs chose to dopt system of ottio bsed o the recommedtios of the Itertiol Orgiztio for Stdrdiztio (ISO). This ottio is used i the emitio ppers for this course

More information

( a n ) converges or diverges.

( a n ) converges or diverges. Chpter Ifiite Series Pge of Sectio E Rtio Test Chpter : Ifiite Series By the ed of this sectio you will be ble to uderstd the proof of the rtio test test series for covergece by pplyig the rtio test pprecite

More information

1 Tangent Line Problem

1 Tangent Line Problem October 9, 018 MAT18 Week Justi Ko 1 Tget Lie Problem Questio: Give the grph of fuctio f, wht is the slope of the curve t the poit, f? Our strteg is to pproimte the slope b limit of sect lies betwee poits,

More information

Numerical Solutions of Fredholm Integral Equations Using Bernstein Polynomials

Numerical Solutions of Fredholm Integral Equations Using Bernstein Polynomials Numericl Solutios of Fredholm Itegrl Equtios Usig erstei Polyomils A. Shiri, M. S. Islm Istitute of Nturl Scieces, Uited Itertiol Uiversity, Dhk-, gldesh Deprtmet of Mthemtics, Uiversity of Dhk, Dhk-,

More information

BRAIN TEASURES INDEFINITE INTEGRATION+DEFINITE INTEGRATION EXERCISE I

BRAIN TEASURES INDEFINITE INTEGRATION+DEFINITE INTEGRATION EXERCISE I EXERCISE I t Q. d Q. 6 6 cos si Q. Q.6 d d Q. d Q. Itegrte cos t d by the substitutio z = + e d e Q.7 cos. l cos si d d Q. cos si si si si b cos Q.9 d Q. si b cos Q. si( ) si( ) d ( ) Q. d cot d d Q. (si

More information

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 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

More information

MATH 104 FINAL SOLUTIONS. 1. (2 points each) Mark each of the following as True or False. No justification is required. y n = x 1 + x x n n

MATH 104 FINAL SOLUTIONS. 1. (2 points each) Mark each of the following as True or False. No justification is required. y n = x 1 + x x n n MATH 04 FINAL SOLUTIONS. ( poits ech) Mrk ech of the followig s True or Flse. No justifictio is required. ) A ubouded sequece c hve o Cuchy subsequece. Flse b) A ifiite uio of Dedekid cuts is Dedekid cut.

More information

MTH 146 Class 16 Notes

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

More information

(200 terms) equals Let f (x) = 1 + x + x 2 + +x 100 = x101 1

(200 terms) equals Let f (x) = 1 + x + x 2 + +x 100 = x101 1 SECTION 5. PGE 78.. DMS: CLCULUS.... 5. 6. CHPTE 5. Sectio 5. pge 78 i + + + INTEGTION Sums d Sigm Nottio j j + + + + + i + + + + i j i i + + + j j + 5 + + j + + 9 + + 7. 5 + 6 + 7 + 8 + 9 9 i i5 8. +

More information

Infinite Series Sequences: terms nth term Listing Terms of a Sequence 2 n recursively defined n+1 Pattern Recognition for Sequences Ex:

Infinite Series Sequences: terms nth term Listing Terms of a Sequence 2 n recursively defined n+1 Pattern Recognition for Sequences Ex: Ifiite Series Sequeces: A sequece i defied s fuctio whose domi is the set of positive itegers. Usully it s esier to deote sequece i subscript form rther th fuctio ottio.,, 3, re the terms of the sequece

More information

King Fahd University of Petroleum & Minerals

King Fahd University of Petroleum & Minerals Kig Fhd Uiversity of Petroleum & Mierls DEPARTMENT OF MATHEMATICAL CIENCE Techicl Report eries TR 434 April 04 A Direct Proof of the Joit Momet Geertig Fuctio of mple Me d Vrice Awr H. Jorder d A. Lrdji

More information

Frequency-domain Characteristics of Discrete-time LTI Systems

Frequency-domain Characteristics of Discrete-time LTI Systems requecy-domi Chrcteristics of Discrete-time LTI Systems Prof. Siripog Potisuk LTI System descriptio Previous bsis fuctio: uit smple or DT impulse The iput sequece is represeted s lier combitio of shifted

More information

SOLUTION OF SYSTEM OF LINEAR EQUATIONS. Lecture 4: (a) Jacobi's method. method (general). (b) Gauss Seidel method.

SOLUTION OF SYSTEM OF LINEAR EQUATIONS. Lecture 4: (a) Jacobi's method. method (general). (b) Gauss Seidel method. SOLUTION OF SYSTEM OF LINEAR EQUATIONS Lecture 4: () Jcobi's method. method (geerl). (b) Guss Seidel method. Jcobi s Method: Crl Gustv Jcob Jcobi (804-85) gve idirect method for fidig the solutio of system

More information

{ } { S n } is monotonically decreasing if Sn

{ } { S n } is monotonically decreasing if Sn Sequece A sequece is fuctio whose domi of defiitio is the set of turl umers. Or it c lso e defied s ordered set. Nottio: A ifiite sequece is deoted s { } S or { S : N } or { S, S, S,...} or simply s {

More information

OBSERVATIONS ON TERNARY QUADRATIC EQUATION z 82 x

OBSERVATIONS ON TERNARY QUADRATIC EQUATION z 82 x Interntionl Reserch Journl of Engineering nd Technology IRJET) e-issn: 395-0056 Volume: 04 Issue: 03 Mr -017 www.irjet.net p-issn: 395-007 OBSERVATIONS ON TERNARY UADRATIC EUATION z 8 x y G. Jnki 1 nd

More information

BC Calculus Review Sheet

BC Calculus Review Sheet BC Clculus Review Sheet Whe you see the words. 1. Fid the re of the ubouded regio represeted by the itegrl (sometimes 1 f ( ) clled horizotl improper itegrl). This is wht you thik of doig.... Fid the re

More information

sin m a d F m d F m h F dy a dy a D h m h m, D a D a c1cosh c3cos 0

sin m a d F m d F m h F dy a dy a D h m h m, D a D a c1cosh c3cos 0 Q1. The free vibrtio of the plte is give by By ssumig h w w D t, si cos w W x y t B t Substitutig the deflectio ito the goverig equtio yields For the plte give, the mode shpe W hs the form h D W W W si

More information

Numerical Solution of Fuzzy Fredholm Integral Equations of the Second Kind using Bernstein Polynomials

Numerical Solution of Fuzzy Fredholm Integral Equations of the Second Kind using Bernstein Polynomials Jourl of Al-Nhri Uiversity Vol.5 (), Mrch,, pp.-9 Sciece Numericl Solutio of Fuzzy Fredholm Itegrl Equtios of the Secod Kid usig Berstei Polyomils Srmd A. Altie Deprtmet of Computer Egieerig d Iformtio

More information

Chapter Real Numbers

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:

More information

SUTCLIFFE S NOTES: CALCULUS 2 SWOKOWSKI S CHAPTER 11

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

More information

Unit 1. Extending the Number System. 2 Jordan School District

Unit 1. Extending the Number System. 2 Jordan School District Uit Etedig the Number System Jord School District Uit Cluster (N.RN. & N.RN.): Etedig Properties of Epoets Cluster : Etedig properties of epoets.. Defie rtiol epoets d eted the properties of iteger epoets

More information

f(bx) dx = f dx = dx l dx f(0) log b x a + l log b a 2ɛ log b a.

f(bx) dx = f dx = dx l dx f(0) log b x a + l log b a 2ɛ log b a. Eercise 5 For y < A < B, we hve B A f fb B d = = A B A f d f d For y ɛ >, there re N > δ >, such tht d The for y < A < δ d B > N, we hve ba f d f A bb f d l By ba A A B A bb ba fb d f d = ba < m{, b}δ

More information

Math 2414 Activity 17 (Due with Final Exam) Determine convergence or divergence of the following alternating series: a 3 5 2n 1 2n 1

Math 2414 Activity 17 (Due with Final Exam) Determine convergence or divergence of the following alternating series: a 3 5 2n 1 2n 1 Mth 44 Activity 7 (Due with Fil Exm) Determie covergece or divergece of the followig ltertig series: l 4 5 6 4 7 8 4 {Hit: Loo t 4 } {Hit: } 5 {Hit: AST or just chec out the prtil sums} {Hit: AST or just

More information

Solution of the exam in TMA4212 Monday 23rd May 2013 Time: 9:00 13:00

Solution of the exam in TMA4212 Monday 23rd May 2013 Time: 9:00 13:00 Norwegi Uiversity of Sciece d Techology Deprtmet of Mthemticl Scieces Cotct durig the exm: Ele Celledoi, tlf. 735 93541 Pge 1 of 7 of the exm i TMA4212 Mody 23rd My 2013 Time: 9:00 13:00 Allowed ids: Approved

More information

Chapter 2 Infinite Series Page 1 of 9

Chapter 2 Infinite Series Page 1 of 9 Chpter Ifiite eries Pge of 9 Chpter : Ifiite eries ectio A Itroductio to Ifiite eries By the ed of this sectio you will be ble to uderstd wht is met by covergece d divergece of ifiite series recogise geometric

More information

For students entering Honors Precalculus Summer Packet

For students entering Honors Precalculus Summer Packet Hoors PreClculus Summer Review For studets eterig Hoors Preclculus Summer Pcket The prolems i this pcket re desiged to help ou review topics from previous mthemtics courses tht re importt to our success

More information

Fast Fourier Transform 1) Legendre s Interpolation 2) Vandermonde Matrix 3) Roots of Unity 4) Polynomial Evaluation

Fast Fourier Transform 1) Legendre s Interpolation 2) Vandermonde Matrix 3) Roots of Unity 4) Polynomial Evaluation Algorithm Desig d Alsis Victor Admchi CS 5-45 Sprig 4 Lecture 3 J 7, 4 Cregie Mello Uiversit Outlie Fst Fourier Trsform ) Legedre s Iterpoltio ) Vdermode Mtri 3) Roots of Uit 4) Polomil Evlutio Guss (777

More information

DETERMINANT. = 0. The expression a 1. is called a determinant of the second order, and is denoted by : y + c 1

DETERMINANT. = 0. The expression a 1. is called a determinant of the second order, and is denoted by : y + c 1 NOD6 (\Dt\04\Kot\J-Advced\SMP\Mths\Uit#0\NG\Prt-\0.Determits\0.Theory.p65. INTRODUCTION : If the equtios x + b 0, x + b 0 re stisfied by the sme vlue of x, the b b 0. The expressio b b is clled determit

More information

1.3 Continuous Functions and Riemann Sums

1.3 Continuous Functions and Riemann Sums mth riem sums, prt 0 Cotiuous Fuctios d Riem Sums I Exmple we sw tht lim Lower() = lim Upper() for the fuctio!! f (x) = + x o [0, ] This is o ccidet It is exmple of the followig theorem THEOREM Let f be

More information

The Definite Integral

The Definite Integral The Defiite Itegrl A Riem sum R S (f) is pproximtio to the re uder fuctio f. The true re uder the fuctio is obtied by tkig the it of better d better pproximtios to the re uder f. Here is the forml defiitio,

More information

Northwest High School s Algebra 2

Northwest High School s Algebra 2 Northwest High School s Algebr Summer Review Pcket 0 DUE August 8, 0 Studet Nme This pcket hs bee desiged to help ou review vrious mthemticl topics tht will be ecessr for our success i Algebr. Istructios:

More information

Chapter System of Equations

Chapter System of Equations hpter 4.5 System of Equtios After redig th chpter, you should be ble to:. setup simulteous lier equtios i mtrix form d vice-vers,. uderstd the cocept of the iverse of mtrix, 3. kow the differece betwee

More information

MA123, Chapter 9: Computing some integrals (pp )

MA123, Chapter 9: Computing some integrals (pp ) MA13, Chpter 9: Computig some itegrls (pp. 189-05) Dte: Chpter Gols: Uderstd how to use bsic summtio formuls to evlute more complex sums. Uderstd how to compute its of rtiol fuctios t ifiity. Uderstd how

More information

Lecture 2. Rational Exponents and Radicals. 36 y. b can be expressed using the. Rational Exponent, thus b. b can be expressed using the

Lecture 2. Rational Exponents and Radicals. 36 y. b can be expressed using the. Rational Exponent, thus b. b can be expressed using the Lecture. Rtiol Epoets d Rdicls Rtiol Epoets d Rdicls Lier Equtios d Iequlities i Oe Vrile Qudrtic Equtios Appedi A6 Nth Root - Defiitio Rtiol Epoets d Rdicls For turl umer, c e epressed usig the r is th

More information

Modified Farey Trees and Pythagorean Triples

Modified Farey Trees and Pythagorean Triples Modified Frey Trees d Pythgore Triples By Shi-ihi Kty Deprtet of Mthetil Siees, Fulty of Itegrted Arts d Siees, The Uiversity of Tokushi, Tokushi 0-0, JAPAN e-il ddress : kty@istokushi-ujp Abstrt I 6,

More information

Chapter 5. The Riemann Integral. 5.1 The Riemann integral Partitions and lower and upper integrals. Note: 1.5 lectures

Chapter 5. The Riemann Integral. 5.1 The Riemann integral Partitions and lower and upper integrals. Note: 1.5 lectures Chpter 5 The Riem Itegrl 5.1 The Riem itegrl Note: 1.5 lectures We ow get to the fudmetl cocept of itegrtio. There is ofte cofusio mog studets of clculus betwee itegrl d tiderivtive. The itegrl is (iformlly)

More information

Chapter 7 Infinite Series

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

More information

Notes on Dirichlet L-functions

Notes on Dirichlet L-functions Notes o Dirichlet L-fuctios Joth Siegel Mrch 29, 24 Cotets Beroulli Numbers d Beroulli Polyomils 2 L-fuctios 5 2. Chrcters............................... 5 2.2 Diriclet Series.............................

More information

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures

FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING. Lectures FACULTY OF MATHEMATICAL STUDIES MATHEMATICS FOR PART I ENGINEERING Lectures MODULE 0 FURTHER CALCULUS II. Sequeces d series. Rolle s theorem d me vlue theorems 3. Tlor s d Mcluri s theorems 4. L Hopitl

More information

,... are the terms of the sequence. If the domain consists of the first n positive integers only, the sequence is a finite sequence.

,... are the terms of the sequence. If the domain consists of the first n positive integers only, the sequence is a finite sequence. Chpter 9 & 0 FITZGERALD MAT 50/5 SECTION 9. Sequece Defiitio A ifiite sequece is fuctio whose domi is the set of positive itegers. The fuctio vlues,,, 4,...,,... re the terms of the sequece. If the domi

More information

2.1.1 Definition The Z-transform of a sequence x [n] is simply defined as (2.1) X re x k re x k r

2.1.1 Definition The Z-transform of a sequence x [n] is simply defined as (2.1) X re x k re x k r Z-Trsforms. INTRODUCTION TO Z-TRANSFORM The Z-trsform is coveiet d vluble tool for represetig, lyig d desigig discrete-time sigls d systems. It plys similr role i discrete-time systems to tht which Lplce

More information

PROGRESSIONS AND SERIES

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.

More information

Convergence rates of approximate sums of Riemann integrals

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

More information

MATRIX ALGEBRA, Systems Linear Equations

MATRIX ALGEBRA, Systems Linear Equations MATRIX ALGEBRA, Systes Lier Equtios Now we chge to the LINEAR ALGEBRA perspective o vectors d trices to reforulte systes of lier equtios. If you fid the discussio i ters of geerl d gets lost i geerlity,

More information

Taylor Polynomials. The Tangent Line. (a, f (a)) and has the same slope as the curve y = f (x) at that point. It is the best

Taylor Polynomials. The Tangent Line. (a, f (a)) and has the same slope as the curve y = f (x) at that point. It is the best Tylor Polyomils Let f () = e d let p() = 1 + + 1 + 1 6 3 Without usig clcultor, evlute f (1) d p(1) Ok, I m still witig With little effort it is possible to evlute p(1) = 1 + 1 + 1 (144) + 6 1 (178) =

More information

(3) (4) The least positive integer solution of (3) is

(3) (4) The least positive integer solution of (3) is Meea K et al.; Sch. J. Phs. Math. Stat., 06; Vol-; Issue- (Feb); pp-5-9 Scholars Joural o Phsics, Mathematics ad Statistics Sch. J. Phs. Math. Stat. 06; ():5-9 Scholars Academic ad Scietiic Publishers

More information

(1 q an+b ). n=0. n=0

(1 q an+b ). n=0. n=0 AN ELEMENTARY DERIVATION OF THE ASYMPTOTICS OF PARTITION FUNCTIONS Diel M Ke Abstrct Let S,b { + b : 0} where is iteger Let P,b deote the umber of prtitios of ito elemets of S,b I prticulr, we hve the

More information

Observation on the Bi-quadratic Equation with Five Unknowns

Observation on the Bi-quadratic Equation with Five Unknowns Interntionl Journl of Reserch in Advent Technolog Vol. No.9 Septemer 018 E-ISSN: 1-97 Aville online t.ijrt.org Oservtion on the Bi-qudrtic Eqution ith Five Unnons 1 1 R.Anuselvi 1 nd N.Ahil * 1 Deprtment

More information

Review of the Riemann Integral

Review of the Riemann Integral Chpter 1 Review of the Riem Itegrl This chpter provides quick review of the bsic properties of the Riem itegrl. 1.0 Itegrls d Riem Sums Defiitio 1.0.1. Let [, b] be fiite, closed itervl. A prtitio P of

More information

PhysicsAndMathsTutor.com

PhysicsAndMathsTutor.com PhysicsAdMthsTutor.com PhysicsAdMthsTutor.com Jue 009 4. Give tht y rsih ( ), > 0, () fid d y d, givig your swer s simplified frctio. () Leve lk () Hece, or otherwise, fid 4 d, 4 [ ( )] givig your swer

More information