Reciprocal Series of K Fibonacci Numbers with Subscripts in Linear Form

Transcription

1 IOSR Joural of Mathematics (IOSR-JM) e-issn: p-issn: X Volume Issue 6 Ver IV (Nov - Dec06) PP wwwiosrjouralsorg Reciprocal Series of K iboacci Numbers with Subscripts i iear orm Sergio alcó Departmet of Mathematics Uiversidad de as Palmas de Gra Caaria Spai Abstract: The aim of this paper is to fid a formula that allows to fid the ifiite sum ofreciprocals of certai k iboacci umbers whose subscripts are i liear formparticularizig this formula we are able to obtai other formulas previously foudby other authors for the case of both classical iboacci ad Pell seueces Keywords:Biet idetity k iboacci umbers k ucas umbers I Itroductio Oe of the more studied seueces is the iboacci seuece [ ] ad it hasbee geeralized i may ways [3] Here we use the followig oe-parametergeeralizatio of the iboacci seuece [45] Defiitio or ay iteger umber 0 ad k k0 k k k k k N k the k iboacci seuece say Note for k = the classical iboacci seuece Pell seuece P P The characteristic euatio of the defiitio is k k is defied recurretly by is obtaied ad for k = it isthe r k r k whose solutiosare 4 that verify k k 4 k 0 0 or the properties of the k-iboacci umbers see [4 5]I particular the Biet Idetity is the covolutio formula k m k k m k k m ially we defie the k-iboacci umbers of egative idex as ( ) k k Defiitio or ay iteger umber k the k ucas seuece [6] say k ad k k0 k k k k k N Note for k = the classical ucas seuece is obtaied 347 ucas seuece P P k k 4 ad ad is defied recurretly by ad for k = it is the Pell- The Biet Idetity for the k-ucas umbers is k The k-ucas umbers are related to the k- iboacci umbers by the relatio k k k II Prelimiary I this sectio we will prove some lemmas that we will eed for the proof of the mai theorem of this paper emma or m () k m k m k m m m m m m m Proof Applyig the Biet formula ( ) k m k m k m DOI: 09790/ wwwiosrjouralsorg 39 Page

2 emma It is () k a4 b k a k a b kb Proof Applyig the covolutio formula ad Reciprocal series of k iboacci umbers with subscripts i liear form ( ) k k the eft Had Side (HS) of this euatio HS k ab b k ab b k ab k b k a bk b k ab k b k a bk b is ( ) k ab k b k a b kb k ab k b k a b kb k a b kb kb k a b kb 3 emma 3 (3) k(a) ( p) k p ka k( a p) Proof Usig the Biet formulas for k ad k ad takig ito accout RHS a a a p a p k 4 is 4a p3 a a p a p a 4a p3 k 4 4a p3 4a p3 p p HS k 4 4 emma 4 k ak b ( k 4) k ak b k a b (4) It is eough to apply the Biet formulas for k ad k 5 emma 5 ( ) a (5) k a b k a k a k a b kb Proof ab ab a a a a ab ab HS k 4 ( ) ( ) ( ) ( ) k 4 ab ab a b a b ab ab a b a b b b a a ( ) ( ) k 4 kb III Mai Result I this sectio we will prove the mai theorem of this paper 3 Mai theorem et us suppose p ad r are iteger umbers The k ( p)( ) (6) j0 k( p)( j) 4 r k p k r k p k( p)( ) r irst we prove the followig euality: Proof We will proceed by iductio o k p r kr If 0 we must prove that ( k 4) k p 4 r k p k p k p r kr k( p)( ) r kr ( k 4) (7) j 0 k ( p )( j ) 4 r k p k p k ( p)( ) r kr romemmawith a pad b r HS ( k 4) RHS k p r k r k r k p r k p r k r k p k p r k r while it DOI: 09790/ wwwiosrjouralsorg 40 Page

3 rom emma 4 with a p r ad b r RHS HS k p r k r Reciprocal series of k iboacci umbers with subscripts i liear form ( k 4) Hece k p r k r k p r k r k p Assumig Mai Theorem is true for we must prove it is true for + If S is the sum of (7) ad a the summads the for + it is S S a a k ( p )( ) r kr ( k 4) Therefore we must prove k p k( p)( ) r k r k( p(4 3) 4 r k p k( p(4 5) 4 r k p k( p)(4 3) 4 r k p k( p)(4 5) 4 r k p ( k 4) ( k 4) k ( p)( 3) r k r k( p)( ) r k r k p k( p)( 3) r k r k( p)( ) r kr k( p)( 3) r k( p )( ) r k p k ( p )( 3) r k( p)( ) r et (p )( ) r be The we must prove k ( p ) k (8) k p k p k 3(p) k p kp k (p) k rom emma 3 with a it is k p k p k k p rom emma with a p ad b p it is k a4 b k a k3( p) k p k p p k p k ( p) k p The i the euatio (8) k ( p) k HS k k p k ( p) k p k k p k ( p) romemma3 k a (p) k a k p k ( pa) with a ad therefore HS k ( p ) k k p k p k p k p O the other had i the euatio (8) it is k ( p) k k k ( p) RHS k p k ( p) k k p k k ( p) rom emma 5 with a c p it is k ( p) k k k ( p) k a ck a k ak a c k ( p) k ( p) k p Takig ito accout k m k mk m fially RHS HS ad the k p k ( p) k k ( p) k formula (7) is prove If a (p )( ) r b r i the euatio (4) the umerator of the RHS of the euatio (6) is ( k 4) ad Mai theorem of this paper is at last prove k( p)( ) r k r k( p)( ) r k r k( p)( ) 3 emma 6 It is verified Proof I [3 4] it is prove Coseuetly lim k k The k lim k 4 k kr k 4 ( k 4) j0 k( p)( j) 4 r k p k p kr (9) lim lim k 4 k k k k k DOI: 09790/ wwwiosrjouralsorg 4 Page

4 Reciprocal series of k iboacci umbers with subscripts i liear form 33 Corollary I this Corollary we will use the euatio (7) or the (6) as appropriate Givig iteger values to r p ad k these formulas are particularized as follows: k ( p)( ) r 0: a b a b j0 k( p)( j) k p k p k( p)( ) p 0: k k k k 4 j0 k j k j0 k j 5 k : [7] k : j0 j P j0 j p : ( ) ( 3 ) ( ) ( 3 ) r : k63 k j0 k 6 j3 k k k k6 3 j0 k 6 j3 k k k k : k : j0 6j3 P j0 6j ( k ) k( p)( ) j0 k( p)( j) 4 k p k p k( p)( ) k k 3 ( k 4) k p 0: j0 k j5 ( k ) k k 3 k k 3 k 5 k : 5 j 3 ( k 4) k 4 j0 k j5 k k 0 j 5 8 k : 8 P 3 j 0 j k k 65 ( k 4) k p : 3 3 j0 k 6 j7 ( k ) ( k )( k 3 k) k 65 ( k 3 k) k6 5 k k 5 k : 5 j Corollary If S ( k r p) ( k 4) 4 3 k j0 k6 j7 ( k ) ( k 3 k) k 0 6 j 7 4 k : P 4 3 j0 6 j7 j 0 k ( p)( j) 4 r k p the or istace: from Corollary S ( 4) S() 9 j0 P8 j3 k k p S( k r p h) S( k r p) k p h 4 S () therefore P j 0 6 j 7 DOI: 09790/ wwwiosrjouralsorg 4 Page

5 35 Corollary 3 kr If p 0 the k 4 ( k 4) j0 kj++4r k ad kr k( r h) k 4 ( k 4) j0 kj++4(r+h) k k( r h) So Reciprocal series of k iboacci umbers with subscripts i liear form k 4 k ( r h) kr adthe k j0 k j4 r k j4( r h) k( r h) kr k 4k ( r ) kr k 4 k ( r) k r k rk ( r ) that is j0 k j4 r k k ( r ) kr j0 k j 4 r k k ( r) k r ially applyig emma 5 with a rad c it is k 4 k (0) k I particular for k ad r 0 it is j0 k j 4 r k( r) kr 5 j 0 j I this form we have foud the formula for the sum of a odd umber of terms of the idicated form while from the formula () (a) of Corollary it is j 0 j rom the differece of these last two formulas ad applyig the Biet formulas we get 4 Refereces [] A Horadam A geeralized iboacci seuece Math Mag 68 (96) [] S Vajda iboacci & ucas Numbers ad the Golde Sectio Theory adapplicatios( Ellis Horwood limited 989) [3] VE Hoggat iboacci ad ucas umbers (Palo Alto CA: Houghto-Miffli 969) [4] S alco A Plaza O the iboacci k-umberschaos Solit &ract 3(5) (007) 65 4 [5] S alco A Plaza The k-iboacci seuece ad the Pascal -triagle Chaos Solit &ract 33() (007) [6] S alco O the k ucas umbers It J Cotemp Math Scieces 6() [7] Robert P Backstrom O Reciprocal Series Related to iboacci Numbers with Subscripts i Arithmetic Progressio The iboacci Quarterly 9() 98 4 DOI: 09790/ wwwiosrjouralsorg 43 Page