Observations on Derived K-Fibonacci and Derived K- Lucas Sequences

Transcription

1 ISSN(Olie): ISSN (Prit): Iteratioal Joural of Iovative Research i Sciece Egieerig ad Techology (A ISO 97: 007 Certified Orgaizatio) Vol. 5 Issue 8 August 06 Observatios o Derived K-iboacci ad Derived K- ucas Sequeces S.Vidhyalakshi M.A.Gopala E.Prealatha Professor Dept. of Matheatics SIGC Trichy Tail Nadu Idia Asst. Professor Dept. of Matheatics Natioal College Trichy Tail Nadu Idia ABSTRACT: I this paper we preset soe rearkable properties of derived k- iboacci ad derived k- ucas ubers. The idetities are ot cosidered earlier. KEYWORDS: Derived k-iboacci sequece ad derived k- ucas sequece Biet s forula. 00 Matheatics Subject Classificatio: B9 B8 I. INTRODUCTION It is well kow that the iboacci sequece is faous for its woderful ad aazig properties. iboacci coposed a uber text i which he did iportat work i uber theory ad the solutio of algebraic equatios. The equatio of rabbit proble posed by iboacci is kow as the first atheatical odel for populatio growth. ro the stateet of rabbit proble the faous iboacci ubers ca be derived. This sequece of iboacci ubers is extreely fruitful ad appears i differet areas i atheatics ad sciece. The iboacci sequece ucas sequece Pell sequece Pell ucas sequece Jacobsthal sequece ad Jacobsthal ucas sequece are ost proiet exaples of recursive sequeces. The iboacci sequece [] is defied by the recurrece relatio k with 0 0. The ucas sequece [] is defied by the recurrece relatio k k k k with 0 The secod order recurrece sequece has bee geeralized i two ways aily first by preservig the iitial coditios ad secod by preservig the recurrece relatio. I this cotext oe ay refer [9]. D.Kala ad R.Mea [] geeralize the iboacci sequece by a b with 0 0 A..Horada[] defied geeralized the iboacci sequece { } H H H with H p H p q where p ad q are arbitrary itegers. B.Sigh O.Sikhwal ad S.Bhatagar [] defied iboacci like sequece by recurrece relatio S k S k S k k with S 0 S.The associated iitial coditios S 0 ad S are the su of the ioacci ad ucas sequeces respectively. i.e S0 0 0 ad S..R.Natividad [] Derivig a forula i solvig iboacci like sequece. He foud issig ters i iboacci like sequece ad solved by stadard forula. V.K.Gupta V.Y.Pawar ad O.Sikhwal [7] defied geeralized iboacci sequeces ad derived its idetities coectio forulae ad other results. V.K.Gupta V.Y.Pawar ad N.Gupta [6] stated ad derived idetities for iboacci like sequece. Also described ad derived coectio forulae ad egatio forulae for iboacci like sequece. B.Sigh V.K.Gupta ad V.Y.Pawar [] preset ay cobiatio of higher powers of iboacci like sequece. Copyright to IJIRSET DOI:0.5680/IJIRSET

2 ISSN(Olie): ISSN (Prit): Iteratioal Joural of Iovative Research i Sciece Egieerig ad Techology (A ISO 97: 007 Certified Orgaizatio) Vol. 5 Issue 8 August 06 The k-iboacci ubers defied by alco Plaza.A [5] depedig oly o oe iteger paraeter k as follows: or ay positive real uber the k-iboacci sequece is defied recurretly by k k k with 0 0 I [0] A.D.Godase ad M.B.Dhake have preseted soe properties of k=iboacci ad k-ucas ubers by usig atrices. I [8] Yashwat K.Pawar G.P.Rathore ad Richa chawla have established soe iterestig properties of k- iboacci like ubers. I this paper we itroduce the derived k- iboacci ad derived k- ucas sequeces. We preset soe properties of derived k- iboacci ad derived k- ucas ubers. Also we illustrate the ethod of obtaiig Diophatie quadruples with property D() ad D() wherei the ebers of the quadruples are represeted by derived k- iboacci ad derived k- ucas ubers. urther we derive telescopig series for derived k- iboacci ad derived k- ucas sequeces. II. METHOD O ANAYSIS Defiitio : Derived k-iboacci sequece or ay positive real uber the derived k-iboacci sequece is defied as 0 0 ad for. Biet for for k is r r where r r r r r r Defiitio : Derived k-ucas sequece or ay positive real uber the derived k-ucas sequece is defied as k 0 ad k for. Biet for for k is r r where r r r r irst 5 derived k-iboacci ubers are give below. k k k k 5 k k 5 6 k 5k k 6 7 k 5k 6k k 6k 0k k k 7k 5k 0k k 8k k 0k 5k k 9k 8k 5k 5k k 0k 6k 56k 5k 6k Copyright to IJIRSET DOI:0.5680/IJIRSET

3 ISSN(Olie): ISSN (Prit): Iteratioal Joural of Iovative Research i Sciece Egieerig ad Techology (A ISO 97: 007 Certified Orgaizatio) Vol. 5 Issue 8 August k k k k k k k k 55k 0k 6k 56k 7k k k 66k 65k 0k 6k 8k irst 5 derived k-ucas ubers are give below. k k k k k k 5 5 k 5k 5k 6 6 k 6k 9k k 7k k 7k k 8k 0k 6k k 9k 7k 0k 9k k 0k 5k 50k 5k k k k 77k 55k k k k 5k k 05k 6k k k k 65k 9 56k 7 8k 5 9k k k k k 77k 0 0k 8 9k 6 96k 9k k 5 k 5k 90k 75k 9 50k 7 78k 5 0k 5 5k Properties of derived k- iboacci ad derived k- ucas sequeces:.. k k k ( ) k k 7. ( ) k 8. k k ( ) k Copyright to IJIRSET DOI:0.5680/IJIRSET

4 ISSN(Olie): ISSN (Prit): Iteratioal Joural of Iovative Research i Sciece Egieerig ad Techology 9. ( ) k k k (A ISO 97: 007 Certified Orgaizatio) Vol. 5 Issue 8 August ( k ) [ ] 6. ( ) k 7. k k ( ) 8 8. ( k ) 9. ( )[ k ] 0. [ ] Costructio of Diophatie quadruples with property D () et a ( k ) b ( k ) be two ubers such that ab [ ] p ( say). Therefore the pair (a b) represets Diophatie -tuple with property D(). et c a b k It is oted that ac [( k ) ] q ( say) bc [( k ) ] r ( say) Thus the triple (a b c) is Diophatie -tuple with property D(). If d is the fourth tuple the by Euler s forula d ( k k k k k k k ) ( ) [( ) ( ) ] ( )( ) Thus the quadruple (a b c d) represets Diophatie quadruple with property D() as the product of ay two ebers of the above set added with uity is a perfect square. I a siilar aer oe ca observe that the quadruple {( k ) ( k ) ( ) ( ) [ ( ) ] ( )( ) k k } is Diophatie quadruple with property D (). Copyright to IJIRSET DOI:0.5680/IJIRSET

5 ISSN(Olie): ISSN (Prit): Iteratioal Joural of Iovative Research i Sciece Egieerig ad Techology (A ISO 97: 007 Certified Orgaizatio) Vol. 5 Issue 8 August 06 Costructio of Diophatie quadruples with property D () et a ( k ) b ( k ) be two ubers such that ab p ( say). Therefore the pair (a b) represets Diophatie -tuple with property D(). et c a b k It is oted that ac [( k ) ] q ( say) bc [( k ) ] r ( say) Thus the triple (a b c) is Diophatie -tuple with property D(). If d is the fourth tuple the by Euler s forula d ( k ) k k k k k ( ) Thus the quadruple (a b c d) represets Diophatie quadruple with property D() as the product of ay two ebers of the above set added with is a perfect square. I a siilar aer oe ca observe that the quadruple {( k ) ( k ) * ( )} is Diophatie quadruple with property D(). Note: Also oe ca observe that the quadruple {( k ) ( k ) k k } is Diophatie quadruple with property D(). III. TEESCOPING SERIES OR DERIVED K- IBONACCI AND DERIVED K- UCAS SEQUENCES: I this sectio we preset theores exhibitig telescopig series for derived k- iboacci ad derived k- ucas sequeces. Theore: If 0 the ) i i ( ) k i ) Replacig by ad by (+) i property () we have ( ) ( ) () which we obtai ) ( ) k ) Replacig by i i () ad takig the suatio fro i= to we have i i) i i k i k i i ( ) i ) ) ) (o expadig) () Copyright to IJIRSET DOI:0.5680/IJIRSET

6 ISSN(Olie): ISSN (Prit): Iteratioal Joural of Iovative Research i Sciece Egieerig ad Techology (A ISO 97: 007 Certified Orgaizatio) Vol. 5 Issue 8 August 06 Theore: If 0 the ) i0 i ( i) ( ) ro () we have ) k k ) ( ) Replacig by i i () ad takig the suatio fro i=0 to we have i) i i k i k i k i k i 0 ( ) 0 ( ) i ) 0 (o expadig) ) 0 ) ( 0 0 ) ( ) () Theore: If 0 the (i) ( ) i ( i ) i ( ) Replacig by ad by (+) i property (0) we have () ( ) ( ) () fro which we get () ) ( ) k ) Multiplyig () ad (5) we have () ) ( ) ) Replacig by i i the above equatio ad takig the suatio fro i= to we have k i k i ( ) i) i k k i k i i k i ( ) i) ) ) (o expadig) (5) Copyright to IJIRSET DOI:0.5680/IJIRSET

7 ISSN(Olie): ISSN (Prit): Iteratioal Joural of Iovative Research i Sciece Egieerig ad Techology (A ISO 97: 007 Certified Orgaizatio) Vol. 5 Issue 8 August 06 ( ) ( ) * ( ) ( ) (6) ( ) Now replacig by (+) by i property (0) ad () we have ( ) ( ) ( ) (7) ad ( ) ( ) (8) Usig (7) ad (8) i (6) we have (i) ( ) i k i k i ( ) ( ) ( ) ( ) Theore: If 0 the (i) ( ) i0 ( i ) i ( ) ro () we have () ) k k ) ( ) Multiplyig () ad (9) we have () ) ( ) ) Replacig by i i the above equatio ad takig the suatio fro i=0 to we have k i ( ) i) i i k k i k i i0 k i 0 ( ) ( ) i ( ) (o expadig) ( ) (9) (0) Theore: 5 If 0 the (i ) [ (i) ] ( ) i0 8 i ( i) ( ) Copyright to IJIRSET DOI:0.5680/IJIRSET

8 ISSN(Olie): ISSN (Prit): Iteratioal Joural of Iovative Research i Sciece Egieerig ad Techology Replace by i i (9) ad () Now ( 9) () (A ISO 97: 007 Certified Orgaizatio) Vol. 5 Issue 8 August 06 (i ) i) i i) i i) i i) i i ( i) = i) i i) i Replace by i i (0) ad ultiplyig the resultig equatio by () we have (i) [ (i) ] i) i 8 i ( i) i) i I the above equatio takig the suatio i=0 to we have k i k i k ( ) [ ( ) ] i) i i k i 8 k i i0 k i 0 ( ) ( ) i ( ) (o expadig) 8( ) IV. CONCUSION () I this paper we have stated ad derived ay properties of derived k- iboacci ad derived k- ucas sequeces through Biet s forulae. Also we have costructed Diophatie quadruples with property D() ad D() where the ubers of the quadruples are derived k- iboacci ad derived k- ucas ubers. ially we preset soe telescopig series ivolvig derived k- iboacci ad derived k- ucas ubers. ACKNOWEDGEMENT The fiacial support fro the UGC New Delhi (-MRP-5/(SERO/UGC) dated arch 0) for a part of this work is gratefully ackowledged. REERENCES [] A..Horada The geeralized iboacci sequeces The Aerica Math.othly 68(5) [] D.Kala ad R.Mea The iboacci ubers-exposed The atheatical Magazie (00). [] T.Koshu iboacci ad lucas ubers with applicatios A.Wileg Itersciece publicatio New Yor 00. [] B.SighV.K.Gupta ad Y.K.Pawar O cobiatios of higher powers of iboacci like sequece Ope Joural of Matheatical odelig ()(0) [5] S.alco Plaza.A O the k- iboacci k- uber Chaos solutios ad ractals 8() [6] V.K.GuptaY.K. Pawar ad N.Gupta Idetities of iboacci like sequece J.Math. Coput.Sci (6) [7] V.K.Gupta Y.K.Pawar ad O.Sikhwal The geeralized iboacci sequeces Theoretical Matheatics ad applicatios ()(0) 5-. [8] Yashwat.K.Pawar G.P.S.Rathore ad Richa Chawla O the k- iboacci like sequece- ike ubers Turkish Joural of Aalysis ad uber theory ()(0) 9-. [9] Yashwat.K.Pawar Bijedra sigh ad V.K.GuptaGeeralized iboacci sequeces ad its properties Palestie joural of atheatics ()(0)-7. [0] A.D.Godase M.B.Dhake O the properties of k- iboacci ad k- ucas ubers IJAAMM()(0) [].R.Natividad Derivig a forula i solvig iboacci- like sequece Iteratioal joural of atheatics Scietific coputig ()(0)9-. [] B.Sigh O.Sikhwal ad S.Bhatagar iboacci- like sequece ad its properties Iteratioal joural of cotep.math. Scieces 5()00) Copyright to IJIRSET DOI:0.5680/IJIRSET