Generalized Fibonacci-Like Sequence and. Fibonacci Sequence

Transcription

1 It. J. Cotep. Math. Scieces, Vol., 04, o., - 4 HIKARI Ltd, Geeralized Fiboacci-Lie Sequece ad Fiboacci Sequece Sajay Hare epartet of Matheatics Govt. Holar Sciee College, Idore, Idia Bijedra Sigh School of Studies i Matheatics Vira Uiversity Ujjai M.P,Idia Shubhraj Pal epartet of Matheatics PMB Gujrati Sciece College, Idore M.P., Idia Copyright 04 Sajay Hare, Bijedra Sigh ad Shubhraj Pal. This is a ope access article distributed uder the Creative Coos Attributio Licese, which perits urestricted use, distributio, ad reproductio i ay ediu, provided the origial wor is properly cited. Abstract Every ter i the Fiboacci Sequece ca be deteried recursively with the help of iitial values F 0 = 0, F =. Siilar is the case with Lucas Sequece. I this paper, we study Geeralized Fiboacci-Lie sequece { } defied by the recurrece relatio = - + -, for all with 0 = ad = +, beig a fixed positive iteger. The associated iitial coditios are the su of ties the iitial coditios of Fiboacci sequece ad the iitial coditios of Lucas sequece respectively. We shall defie Biet's forula ad geeratig fuctio of Geeralized Fiboacci-Lie sequece. Maily, Iductio ethod

2 Sajay Hare, Bijedra Sigh ad Shubhraj Pal ad Biet's forula will be used to establish properties of Geeralized Fiboacci-Lie sequece. Matheatics Subject classificatio: B, B7, B Keywords: sequece Fiboacci sequece, Lucas Sequece, Geeralized Fiboacci-Lie. Itroductio The sequece of Fiboacci ubers {F } is defied by F = F - + F -,, F 0 = 0, F =.. The sequece of Lucas ubers {L } is defied by L = L - + L -,, L 0 =, L =.. The Biet's forula for Fiboacci sequece is give by F. where Golde ratio. 8 ad Siilarly, the Biet's forula for Lucas sequece is give by L.4 Fiboacci sequece have bee geeralized by ay ways; soe chagig the recurrece relatios while preservig the iitial ters, soe alterig the iitial ters but aitaiig the recurrece relatios [,,0 ]. I this paper, we preset various properties of the Geeralized Fiboacci-Lie sequece { } defied by = - + -, for all. with 0 = ad = +, beig a fixed positive iteger

3 Geeralized Fiboacci-lie sequece 7 Here the iitial coditios 0 ad are the su of ties the iitial coditios of Fiboacci sequece ad the iitial coditios of Lucas sequece respectively. i.e. 0 = F 0 +L 0, = F + L. The few ters of the sequece {M } are, +, +, 4+, 7 +, +, ad so o.. Preliiary Results of Geeralized Fiboacci-Lie sequece First we itroduce soe basic results of Geeralized Fiboacci-Lie sequece ad Fiboacci Sequece. The relatio betwee Fiboacci Sequece ad Geeralized Fiboacci-Lie sequece ca be writte as = F + L, o.. The recurrece relatio. has the characteristic equatio x - x - = 0 which has two roots ad. Now otice a few thigs about α ad β: α + β =, α - β = ad αβ = -.. usig these two roots, we obtai Biet's forula of recurrece relatio. The geeratig fuctio of { } is defied as x x.4 x x 0. Properties of Geeralized Fiboacci-Lie Sequece Geeralized Fiboacci-Lie sequece { } has ay fasciatig properties [,,,8]

4 8 Sajay Hare, Bijedra Sigh ad Shubhraj Pal Sus of Geeralized Fiboacci-Lie ters: Theore.. Su of first ters of the Geeralized Fiboacci-Lie sequece{ } is.... This idetity becoes.... Theore.. Su of the first ters with odd idices is.... Theore.. Su of the first ters with eve idices is Theore. to. ca be proved by Matheatical iductio. If we subtract equatio.4 ter wise fro equatio., we get alteratig su of first ubers = = Addig + to both sides of equatio., we get = = + -. Cobiig. ad., we obtai = Theore.4 Su of the squares of first ters of the Geeralized Fiboacci-Lie Sequece is....8

5 Geeralized Fiboacci-lie sequece Now we state ad prove soe idetities for sequece { }.Soe of the is siilar to Fiboacci ad Lucas sequeces [4,7, ] Theore.. For every iteger 0, =. Theore.. For every positive iteger, = + - -,.0 Theore.7. For every positive iteger, = Proof. we shall use Matheatical iductio over. It is easy to see that for =, 0 - = = -, which is true. Assue that the result is true for =. The. Addig + to each side of equatio., we get Which is precisely our idetity whe = +. Therefore, the result is true for = + also.

6 40 Sajay Hare, Bijedra Sigh ad Shubhraj Pal Hece,. Theore.. For every positive iteger, /.... Proof. By usig Biet's forula, we have

7 Geeralized Fiboacci-lie sequece 4 REFERENCES [] A. F. Horado, Basics Properties of Certai Geeralized Sequece of ubers, The Fiboacci Quarterly,,-7. [] A. T. Bejai, ad J.J Qui, Recoutig Fiboacci ad Lucas idetities, College Math, J.,0, No., -. [] B. Sigh, O. Sihwal, S. Bhatagar, Fiboacci-Lie Sequece ad its properties, It. J. Cotep Math. Scieces, Vol., 00, No.8, [4]. M. Burto, Eleetary Nuber Theory, Tata McGraw Hill, Publishig Copay Ltd., New elhi, 00. [] J.Z. Lee, ad J.S.Lee, Soe Properties of Geeralizatio of the Fiboacci Sequeces, The Fiboacci Quarterly, 87,No., 0-7. [] L. Carlitz, A ote o Fiboacci Nubers, The Fiboacci Quarterly, 4, - 8. [7] N. N. Vorobyov, The Fiboacci Nubers,.C. Health Copay, Bosto,. [8] S. L. Basi, ad V.E. Hoggatt, A Prier o the Fiboacci Sequece, Part II, The Fiboacci Quarterly,, -8. [] V. S. Harris, O Idetities ivolvig Fiboacci Nubers, The Fiboacci Quarterly, Vol., 4-8. [0] V. H. Badshah, M.S.Teeth, M.M. ar, Geeralized Fiboacci-Lie Sequece ad its properties, It. J. Cotep Math. Scieces, Vol.7, 0, No.4, -4. Received: February 8, 04