Cubo A Mathema tical Jo1wnal \'ol.05/Nf103 - OCTOBER 2003 Proofs for the Limit of R a tios o f Consec utive Te rms in Fibo nacci Seque nce C /1110-Piug C he11 Dcp"rtr11e11t of tl pplitd M" themat11.:s m1d Jnfom1atlcs. Jwo;uo fris t1t11te of Tedm ology. Jwoz uo C:ity, f fo11a11 454000. Clwia A i-Qi Liu Smu11e11Xi.a Collegc o/ Voc(l f101wf Tedmology. S"mnena:ia Oit y, He11 afl 472000, Clww (lll(/ Fe ug Q i Depa rt me11t of A pplicd Matl1wwt1cs a11d lnformatics. Jiaowo l mtit rd e o/ Teclmology, JiaozlllJ Cit y, llenan 454000, Clima AnSTll AC'T. In thc short note, six proofs fo r l hl• limit of rnt ios of consecu tivc tcrms in F' ibonacci .sequcncc nrc providcd , includ ing usi ng Wcicrst rass.Bolzano theorcm , series nll'thod . by dl'fin it ion oí lim it , diffcrcncc cquat ion mcthod, matrix mcthod , a lgebraic 11wtl1U(I. :moo 1\/n1h,-,,.n11r• S ub1u l Cln.t~tjicntion. Primnry 11839. J..·y u'Onl• rrnd phrcur:s. llnlio, Fibonncci numbcto 1 Fund o f .J111ozuo l n~1itul f' o í Tcchm1IOll.,\'. ('h1111• 211 L.,imit o( 1J1e Ra tios of Co11secutive Ter.ms in Fibonacci Sequence Introd uct ion lt is well -kn owu that t he Fibonacci sequence {F,,};;",,, 1 is defined by the following recurren ce formul a { Fu+2 = Fn+t +F." ' F1=1 , F2 = 1 We al so cal\ Fn t he F ibonacci number. Its general term can be expressed as _ 1 [(l+,/5)" (I-./5)"] F,,-Js -,- - -,- for 11 EN. (1) (2) Th e F ih onncci mnnbe rs give the numbe r of pairs of rabbits n mont hs after a singl e pnir begi 11 s br ecd in g (a nd ue wly born bunnies are a.ssumed to begin breeding when they are two 111 0 111 h!' o ld ). Defin e F .. +1 X,.=---¡;:- , (3) t he n t he seq ue nce { :z:., } ~= I converges , this limit (•1) i!' ca ll cd Go ld e n Rat io. Thc rnt ios o f nltcmat.c Fibouacci umuben; are givcu by the conve rgeut.s t.o efi - 2 uml urc said to mensure t he frnction of a tum between successive lea.ves 011 the ·sta\k of a plaut ( Phy ll ot a.xis): 4 fo r elm and linden, & for beech aud hazel , ~ for oak a.nd apple , ~ for poplar nu d rosf'. fJ fa r wil! ow and 11.lmoncl, etc .. The Fibonacci numbers are sometimes call ed Pinc C'u 111 · Nuru hcrs. The ro le of t.h e Fibouacci muuhers iu botany is som etimes callcd Ludwig 's l. nw . To prove t hc convc rgcuce all(] t.o salve its limit, oí the sequeuce {x.,} ~:= l is ll stnudnrd 1•x1• 1-ci~· or cxmu plC' i11 calc nlus aud 111111,hematical una!ysi:-; íor grnc.hmt c stu{'l"i l's 1111•! hod. b,\" 1101 11101101ouic . Using fonnu la (5) agaiu giw:s lb .i:,.- 2 = l + :c.,l-1 = 1 + 1 +! t, = 2it·: : .. ! . (7) 2:i:,, + l 2x,, _2 + 1 .e,. - .Cn-2 .c,._2 - i · ., = ~ - l + :i:,.- 2 = (.c.,+ i )(.c,,_2 + ! )' (8) Sincc (x ,, + 1)(.c11 _ 2 + J) > O, t hen (x,,+2 - :i.: 11 )(:1:,, - x,, _2) ?: O. This implies that the scq uences {.i:2,. _ 1 ):C= i a nd (x2 ,, }i.'::: i are al! monotonic. In fact , by iu duct ion, we can prove l b a t {:i.:2,,- d i,= 1 is increasi11g a n.: 1_ 1 !x.¡ - x2J, lx2k-4-l -x2k-1I S 4k. 1_, lx3-x1I, (12) (13) where k E N. By propertoies of series wi th positive terms , it is deduced thato the sePies L;;o=1 (x2 k+2 - x2k) and ¿:;;o= 1 (x2>.:+1 - X2k·-d a11e all absolutely convergent. Since ' Xu· = X2 + ¿(x2; - X2(i-1j), (14) i=2 ' X2k-I =X¡+ ¿(X2i-l - X2;-:i) , (15) i=2 Uhc sequcuccs {x2d f'= 1 and {.t:2l·-dr.; 1 c011verige. The rcst is sam e as t.hat in the fi.rst 1~ro0f. o The third proof: by definition af limit. Note tihat the equation x = l + ~ corresponding to .t,.+ 1 = 1 + -!_: for n EN has unique positiive root ~· By defü1itio11 of limit, using the result l ::; x .. ::; 2 for n E N, we have thCH l. 1+v'5 1 (J5 - l)"-'1·· l+v'51-(J5-l)" O .i:,.--2- ~ -2- :J,¡-~ - ~ - (17) n~" - ce. Tl1crf'fon• li n1,,_«>:r., = 1±F· o The fo nrlh proo f: diffc r e nc e equaL ion method. Fr1>HI v .. - ~ = ¡. ~·- 1 f r ... \\' I ' nl11111 11 CJrno-Pi11g C J1e11 , Ai·Qi Uu & Feng Qi bhc linear differencc cquat.ion o f second order with constnnt cocfficients: Fn+2 - F .. + 1 - F., =O. ( 18) lts cigen<•quut.iou is ,\2 - ,\ - 1 =O. it.s e igenvalues ni-e ,, = 1 + J5 '" = 1 - J5 2 1 • 2 The gencrnl solutio11 of t.hc diffcrence equntio n is - (l+ JS)" (' -JS)" F,, - C1 - 2- +C, - 2 - . By the iu it inl condi t.i011s F1 = l llltd Fi = l. we o bt nin Heuce F =_!__ [(1 +J5)" - (1 -J5) "] " J5 2 2 nml [1.±..>d] .... - [1=1] "+' F .. + 1 2 2 _ 1 + /5 x,,=T.= r~r- r~r 2 o The fiíth p rooí: matrix m cthod. I!: is ~ns.v t.o Sf'e that (f~·-') = (' ') (F•.•+ •) = (' 1)" (F') = (¡ 1)" (¡) (lg) l 11_ 1 1 O F11 l O Fi 1 O 1 for 11 E N. Ll't A= (: fi). thc11 t he eigcuvalucs o f t hc squnre mntrix A are ,\1 = ~ aud ,\1 = 1-=:P. t lu·ir corrcsponding eigc11vect0rs are ( '=f!i ) ( 1=/f' ) 111 = . 0 2 = 1 1 (20) 27 28 Umit of rlie Rnr.ios of Co11secutive Terms in Fibonacci Sequence Tnking ( !.±>d ~) P - , , - ' 1 1 t hcn ~his imp\ies A"= p 2 " p- l ([!±.i!J" o ) o ['-=f'] , , 1 ( [1±.il]""' - [~]"" ~ 7s [ "I']" - [ '-=f']" ~[~r+l -~[~r+l) ~[~]"-~[~]" . tll Ld Thus - __!__ [(l + v's)"" - (1 - v's)""] F,.+ 1 - J5 2 2 ' thnt is . ~ __!__ [( l + v's)" - (1 - v's)"] f,, v's 2 2 for 11 EN. Tht• rcsl is snme ns ~hnt, in the tlhird proof. The sixth proof: algebrnic met hod. Le t us rewri!.c F .. +2 = F11 - 1 + F,. ns tlm! i:- (21) (22) (23) (25) (2G) o Clwo-Pi11g C /ie11. Ai-Qi Liu & Fe11g Qi 29 Lc1 { p+q = l. /)(/= - l. (29) Lhcn wP obtam solu11011~ of (29) as follows { p= Ll;f',