Convergence of Quotients of Consecutive Terms of a Generalized Secondary Fibonacci Sequence

Transcription

1 Intenational Mathematical Foum, Vol. 6, 2011, no. 40, Convegence of Quotients of Consecutive Tems of a Genealized Seconday Fibonacci Seuence Loenzo J. Matínez Univesidad de Caldas loenzo.matinez h@ucaldas.edu.co Alvao H. Salas Univesidad de Caldas, Manizales, Colombia Depatment of Mathematics Univesidad Nacional de Colombia asalash2002@yahoo.com FIZMAKO Reseach Goup Abstact In this pape we conside a genealized Seconday Fibonacci seuence. We study the convegence of the seuence of uotients of its consecutive tems and we give an estimation fo the speed of its convegence. Keywods and phases: Fibonacci numbes, seconday Fibonacci seuence, speed of convegence, metallic means family 1 Intoduction Some numbes ae well known because of thei mathematical popeties, applications and pesence in diffeent aeas, e.g., φ = 1+ 5 and θ = These constants ae called the golden mean and the silve mean, espectively [12, 7, 5, 9]. φ and θ have a lot of geometic, algebaic and analytical mathematical popeties which ae useful in the study of achitectual popotion, at, compute science, and gowth of some biological systems [6, 3, 10, 17, 15]. Analogously to the way the golden mean is elated to the Fibonacci Seuence, the silve mean is elated to the Pell Seuence [9]. These seuences ae defined ecusively by:

2 1956 L. J. Matínez and A. H. Salas and 0 if n =1 F n = 1 if n =2 F + F n 2 if n 3 0 if n =1 P n = 1 if n =2 2P + P n 2 if n 3 (1) (2) espectively. The theoy and genealization of Fibonacci Numbes and the golden mean is an impotant banch of moden mathematics [20, 15]. Accoding to [18], ϕ and θ ae membes of a vey special goup of uadatic positive iational numbes known as the Metallic Means Family (MMF) and aise as positive solution of the euations x 2 px =0;p, Z +. (3) With appopiate values fo p and in (3) a membe of the Metallic Means Family can be obtained, i.e., the golden mean, by setting p = 1 and =1; the silve mean, with p = 2 and = 1; the coope mean by setting p = 1 and = 2 and so on[17]. The Fibonacci and Pell seuences ae paticula cases of the genealized seconday Fibonacci seuence (GSFS). They satisfy elations of the type a if n =1 c n = b if n =2 (4) pc + c n 2 if n 3; p, Z + Theoem 1. Suppose that a, b, p, > 0 ae positive numbes such that b a def = μ def = p + p (5) 2 Then the seuence {a n } n=1 defined by whee {c n } n=1 is defined by (4) conveges and a n = c n+1 c n, (6) lim a n = p + p2 +4. (7) n 2

3 Convegence of uotients of consecutive tems 1957 Poof. Let a, b, p, > 0. Suppose that >μ. (8) The case when <μmay be consideed analogously. Fist at all, c n > 0 fo all n =1, 2, 3,... Indeed, c 1 = a>0 and c 2 = b>0. Suppose that a j > 0 fo j =2,..., k. Then c k+1 = pc k + c k 1 >pc k > 0. Now, obseve that 0 <a n = c n+1 c n = pc n + c c n = p + c c n = p + a fo n =2, 3,... (9) It follows fom (9) that a n = p + p + a n 2 fo n =3, 4,... (10) In paticula, a n+2 a n = ( + pa n )( + pa n 2 ) (a n a n 2 ) fo n =3, 4, 5,... (11) It is clea that μ is the positive oot of the euation paticula, x 2 px =0. In μ 2 = pμ + and since >μ, 2 p >0. (12) We now will show that the subseuence {a 2n } n=1 of even tems is inceasing and the subseuence {a 2 } n=1 of odd tems is deceasing and a 2m μ a 2 fo any m, n =1, 2, 3,... (13) Indeed, obseve that a 1 = c 2 c 1 = b a = and fom (10) and (12) we obtain a 3 a 1 = p + p + = p 2 p + p < 0. (14) On the othe hand, fom (9) a 4 a 2 = p + ) (p + a1 a 3 = (a 1 a 3 ) a 1 a 3 > 0. (15)

4 1958 L. J. Matínez and A. H. Salas Then 0 <a 3 <a 1 and a 4 >a 2 > 0. (16) We now poceed by induction. Suppose that a 2k+1 <a 2k 1 and a 2k+2 >a 2k fo some k 1. Making use of (11) gives a 2k+3 a 2k+1 = a 2k+1 a 2k 1 ( + pa 2k+1 )( + pa 2k 1 ) < 0 (17) and a 2k+4 a 2k+2 = a 2k+2 a 2k ( + pa 2k+2 )( + pa 2k ) > 0 (18) By the pinciple of mathematical induction, a 2n+1 <a 2 and a 2n+2 >a 2n fo all n =1, 2, 3,... (19) We now will show that a 2 >μand a 2n <μfo all n 1. (20) We again poceed inductively. In view of (8) we have μ > and then a 2 μ = p + μ>p+ = 1 (2 p ) < 0 and a 1 μ = μ>0. We have poved that a 1 >μand a 2 <μ. Suppose that a 2k 1 >μand a 2k <μfo some k 1. Then, by vitue of (11), a 2k+1 μ = p + p + μ>p+ a 2k 1 p + μ μ = p μ2 pμ + pμ =0

5 Convegence of uotients of consecutive tems 1959 and a 2k+2 μ = p + p + a 2k μ< p + μ μ = p μ2 pμ + pμ =0. We have poved (19) and (20), that is, a 2 <a 4 <a 6 < <μ< <a 5 <a 3 <a 1. By the Weiestass theoem, thee exist 0 <α= lim n a 2n and β = lim n a 2. We have ( α = lim a 2n+2 = lim n n p + ) p + = p + a 2n p + α fom whee p α2 pα + pμ =0 and then α = μ. In a simila way, β = μ. This implies that lim n a n = μ. Indeed, let be ε>0 any positive numbe. Thee exist two positive intege numbes N 1 and N 2 such that a 2n μ <εfo n>n 1 and a 2 μ <εfo n>n 2. Let N = max (2N 1, 2N 2 ) and let k > N. If k is even, say k = 2n then 2n >N 2N 1, n>n 1 and then a k μ <ε.ifkis odd, say k =2 then 2n 1 >N 2N 2, n>n 2 +1/2 >N 2 and then a k μ <ε. This poves (7). Remak. If <μthen we may show that the subseuence of even tems of the seuence {a n } n=1 is deceasing, while the subseuence of odd tems of this seuence is inceasing and a 1 <a 3 <a 5 < <μ< <a 6 <a 4 <a 2. This happens, fo example, if a = b = p = = 1 (Fibonacci seuence) and a = b =1,p =2, = 1 (Pell seuence). Thus, fo the Fibonacci seuence, =1<μ=

6 1960 L. J. Matínez and A. H. Salas and fo the Pell seuence, =1<μ=1+ 2. On the othe hand, if = μ we may show that {a n } n=1 is a constant seuence and each of its tems euals μ. Then {a n } n=1 conveges to μ. Thus, in any case, a n μ (n ). We now poceed to estimate the speed of convegence of the seuence {a n } n=1.we have the following Theoem. Let C = max{ a 3 a 1, a 4 a 2 }. If μ thee exists a constant ρ such that 0 <ρ<1 fo which a n μ < Cρ 1 ρ fo any n =1, 2, 3,... (21) Poof. Suppose that >μ(the case <μis simila). Making use of (17), (18) and (20) we obtain a 2k+3 a 2k+1 = < = ( + pa 2k+1 )( + pa 2k 1 ) a 2k+1 a 2k 1 ( + pμ)( + pμ) a 2k+1 a 2k 1 ( + pμ) 2 a 2k+1 a 2k 1. We have poved that whee a 2k+3 a 2k+1 <ρ 1 a 2k+1 a 2k 1 fo any k 1, (22) ρ 1 = 2 (0, 1). (23) ( + pμ) On the othe hand, since a 2k 1 <a 1 = and a 2k 1 >μ, Similaly, a 2k = p + a 2k+2 = p + = pa 2k 1 + a 2k 1 a 2k 1 = pa 2k+1 + a 2k+1 a 2k+1 > pμ +. (24) > pμ +. (25)

7 Convegence of uotients of consecutive tems 1961 We have a 2k+4 a 2k+2 = < = = a 2k+2 a 2k ( + pa 2k+2 )( + pa 2k ) a 2k+2 a 2k ( )( ) p(pμ + ) p(pμ + ) + + p 2 ( + p 2 μ + p) 2 a 2k+2 a 2k ( ) 2 a 2k+2 a 2k. + pμ2 We have poved that whee a 2k+4 a 2k+2 <ρ 2 a 2k+2 a 2k fo any k 1, (26) ρ 2 = ( + pμ2 ) 2 (0, 1). (27) Let A n = a 2 and B n = a 2n (n =1, 2, 3,...). Taking into account (22) we obtain fo k 1 a 5 a 3 < ρ 1 a 3 a 1 a 7 a 5 < ρ 1 a 5 a 3. a 2k 1 a 2k 3 < ρ 1 a 2k 3 a 2k 5 a 2k+1 a 2k 1 < ρ 1 a 2k 1 a 2k 3 Multiplying tem by tem these ineualities yields A k+1 A k = a 2k+1 a 2k 1 <ρ k 1 1 a 3 a 1 = ρ k 1 1 C 1, whee C 1 = a 3 a 1. (28) In a simila way, B k+1 B k = a 2k+2 a 2k <ρ k 1 2 a 4 a 2 = ρ k 1 2 C 2, whee C 2 = a 4 a 2. (29)

8 1962 L. J. Matínez and A. H. Salas Ineualities (28) and (29) hold fo any k 1. Let m>n. We have A m A n < < < m 1 A k+1 A k k=n m 1 k=n k=n ρ k 1 1 C 1 1 C 1 = C 1ρ1. 1 ρ 1 ρ k 1 We thus have A n A m < C 1ρ1 fo any m>n. 1 ρ 1 Letting m in this last ineuality gives Similaly, Finally, obseve that a 2 μ C 1ρ1 fo any n 1. (30) 1 ρ 1 a 2n μ C 2ρ2 fo any n 1 (31) 1 ρ 2 ρ 2 = ( ) 2 > + pμ2 ( ) 2 = + pμ2 μ ( + pμ) 2 = ρ 1 (32) and then It is clea fom (30), (31) and (33) that C 2 ρ2 > C 2ρ1 (33) 1 ρ 2 1 ρ 1 a n μ < Cρ 1 ρ, (34) whee C = max{c 1,C 2 } and ρ = ρ 2 = max{ρ 1,ρ 2 }. This poves the theoem.

9 Convegence of uotients of consecutive tems Conclusions We genealized some esults concening the Fibonacci and Pell numbes. We did not make use of the theoem about the existence and uniueness of the solution of a geneal linea ecuence seuence of second ode. The main esult in this wok, i.e. Theoem 1, can also be obtained diectly by using Binnet s Fomula [20],[3]. Refeences [1] Apostol, T. Calculus volumen I. Editoial Reveté [2] Black, J., Geen A. Gods, demons and symbols of the ancient Mesopotamia. An illustated dictionay. Texas Univesity Pess [3] Dunlap R. The Golden Ratio and Fibonacci Numbes. Wold Scientific Publishing [4] Fitz, K. The discovey of incommensuability by Hippasus of Metapontum, annals of Math., 46 (1945) [5] Ghyka, M. Estética de las popociones en la natualeza y en las ates [6] Ghyka, M. El númeo de oo. Ritos y itmos pitagóicos en el desaollo de la civilización occidental. Editoial Poseidón [7] Ghyka, M. The geomety of at and life. Dove Publications, Inc [8] Leonado y la matemática. Coopeativa Editoial magisteio [9] Kappaff, J., Connections: The Geometic Bidge between At and Science, 2nd ed., Singapoe: Wold Scientific Publ. (2001). [10] Kappaff, J., Beyond Measue: A Guided Tou though Natue, Myth, and Numbe, Singapoe: Wold Scientific Publ. (2002). [11] Withfod, E. The Pell Euation. Pess of The New Ea Pinting Company Lancaste. PA [12] Livio, M. The Golden Ratio: The Stoy of Phi, the Wold s Most Astonishing Numbe. Random Hause Inc [13] Vooviev, N. N., Suite de Fibonacci, Mi, Moscou, [14] Newman, J. El mundo de las matemáticas. Colección Sigma. Ediciones Gijalbo S. A.

10 1964 L. J. Matínez and A. H. Salas [15] Stakhov, A., Rozin, B. Theoy of Binet fo Fibonacci and Lucas p- numbes. Disponible en linea en [16] Spivak, M. Calculos, Cálculo infinitesimal. Editoial Reveté, s.a [17] Vea W. de Spinadel,THE FAMILY OF METALLIC MEANS Cento de Matemática y Diseño -Facultad de Auitectua, Diseño y Ubanismo. Univesidad de Buenos Aies. José Maía Paz 1131 Floida (1602) Buenos Aies Agentina [18] Spinadel, Vea. The Family of Metallic Means. Disponible en http/ [19] Vinogadov, I. Fundamentos de la teoía de los númeos. Editoial Mi [20] Vooviev, N. N., Suite de Fibonacci, Mi, Moscou, Received: Febuay, 2011