New Results for the Fibonacci Sequence Using Binet s Formula

Transcription

1 Iteratioal Mathematical Forum, Vol. 3, 208, o. 6, HIKARI Ltd, New Results for the Fiboacci Sequece Usig Biet s Formula Reza Farhadia Departmet of Statistics, Loresta Uiversity Khorramabad, Ira Rafael Jakimczuk Divisió Matemática, Uiversidad Nacioal de Lujá Bueos Aires, Argetia Copyright c 208 Reza Farhadia ad Rafael Jakimczuk. This article is distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. Abstract Let k ad h = 0,..., k. I this ote we study the mootoicity of the sequeces F k+h /, where F deotes the -th Fiboacci umber. I particular, we prove that the sequeces F 2 / ad F 2+ / are strictly icreasig for. We also obtai asymptotic formulae for k= log F k, k= F k ad prove some its. Mathematics Subject Classificatio: B39, B99 Keywords: Sequeces, mootoicity, Fiboacci umbers Itroductio ad Preiary Notes I 202, the Italia mathematicia Leoardo Fiboacci 7-2, through a problem rabbit problem i his book Liber Abaci, he itroduced the followig sequece of umbers: 0,,, 2, 3,, 8, 3, 2, 34,, 89, 44, 233,...

2 260 Reza Farhadia ad Rafael Jakimczuk This sequece is called the Fiboacci sequece, ad its terms are kow as Fiboacci umbers. The Fiboacci sequece has a simple rule. I fact, startig with 0 ad, every ext umber is foud by addig up the two umbers before it. I mathematical terms, if F be the -th Fiboacci umber, the F = F + F 2. with F 0 = 0 ad F =. There are may methods ad explicit formulas to fidig the -th Fiboacci umber. For example, the well-kow Biet s formula discovered by the Frech mathematicia Jacques Philippe Marie Biet i 843 states that: F = [ + 2 The Biet s formula ca also be writte as ]. 2 F = ϕ ϕ, where ϕ = , is the golde ratio. 2 There are may papers ad books o the Fiboacci umbers. See, for example, []. I this article, we are iterested i studyig the mootoicity of sequeces cosistig of Fiboacci umbers ad also i asymptotic formulae ad its where the Fiboacci Numbers appear. 2 Mai Results Lemma 2. The followig it holds log + x x 0 x =. Proof. Use L Hospital s rule. The lemma is proved. Theorem 2.2 Let k ad h = 0,..., k. If either h = 0 or h = the the sequece F k+h is strictly icreasig from a certai value of ad its it is ϕ k. O the other had, if h 2 the the sequece F k+h is strictly decreasig from a certai value of ad its it is ϕ k.

3 Results for the Fiboacci sequece usig Biet s formula 26 Proof. We have Biet s formula F k+h ϕ k+h k+h ϕ = ϕ = ϕ k h + k+h+ ϕ 2h = exp k log ϕ + log a + log = exp k log ϕ + log a + = ϕ k a + c fc b b + c b, 2 where a = ϕh, c = k+h+ = O, b = > ad f by Lemma ϕ 2h 2.. Note that a immediate cosequece of the secod lie of equatio 2 is the it of the sequece is ϕ k. Let us cosider the sequece see the last lie of equatio 2 We have A = k log ϕ + log a + fc b A + A = log a + + O = b = k log ϕ + log a + O. b log a + O + + b. Therefore, from a certai value of the sig of A + A is the sig of log a = log ϕ h. If either h = 0 or h = the log a > 0 ad cosequetly the sequece A is strictly icreasig from a certai value of. O the other had, if h 2 the log a < 0 ad cosequetly the sequece A is strictly decreasig from a certai value of. The theorem is proved. I the case k = 2 we ca to prove that both sequeces are strictly icreasig from =. Lemma 2.3 Let a >.284 is a costat. The for every x > 0 we have 2x a 2x loga + loga a 2x > a 2x a 2x loga2x a Proof. Let us cosider the real fuctio hx = a 2x. The derivative of a 2x hx is h x = 2 logaa4x + that is positive for a > ad every real umber a 2x x, which meas that the fuctio hx with a > is strictly icreasig. I 2x.

4 262 Reza Farhadia ad Rafael Jakimczuk additio by calculatio we have hx > for a.284 ad x > 0. Now let s cotiue the proof. We have 2xa 2x loga = a 2x loga 2x > a 2x loga 2x a 2x = a2x loghx. We have loghx > 0 for a.284 ad x > 0 sice hx >. Hece, we have 2xa 2x loga a 2x loga 2x > 0, a.284, x > 0. 3 a2x 2x loga Sice ad loga 2x both are positive for a > ad x > 0, a 2x a 2x a 2x by addig these to the left side of iequality 3, we have 2xa 2x loga a 2x a 2x loga2x loga +2x > 0, a2x a 2x x > 0, a.284. Cosequetly 2x a 2x loga + loga > a 2x a 2x a 2x loga2x, a2x x > 0, a.284. The lemma is proved. Theorem 2.4 The sequece F 2 is strictly icreasig for. Proof. Usig Biet s formula we have F 2 = ϕ2 ϕ 2 = ϕ2 ϕ 2. 4 Let us cosider the followig cotiuous fuctio correspodig to 4: fx = ϕ2x ϕ 2x x, x > 0. We obtai the derivative of fx as follows: /x ϕ 2x f ϕ x = 2x x 2 2xϕ2x logϕ + logϕ ϕ 2x x ϕ 2x log ϕ 2x ϕ ϕ 2x 2x + log /x ϕ 2x ϕ 2x 2x 2. /x Usig Lemma 2.3 we kow that f x > 0 for x > 0. Sice the fuctio fx has a positive derivative for x > 0, so fx is strictly icreasig for every x > 0, cosequetly the sequece F 2 is strictly icreasig. The theorem is proved.

5 Results for the Fiboacci sequece usig Biet s formula 263 Lemma 2. If x > 0 the the followig iequality holds 0 < log + x x <. Proof. The fuctio fx = x log + x has positive derivative for x > 0 ad f0 = 0. The lemma is proved. Theorem 2.6 The sequece F 2+ is strictly icreasig for. Proof. If k = 2 ad h = equatio 2 gives F 2+ = exp 2 log ϕ + ϕ log + f ϕ 2 ϕ 4 where, by Lemma 2., we have 0 < f <. Let us cosider the sequece see A = 2 log ϕ + log ϕ + f ϕ 2 ϕ 4. We have where A + A = F = f + ϕ 2 ϕ log + F, 6 + ϕ 4 + f + ϕ 2 ϕ 4, ad cosequetly F ϕ 2 ϕ ϕ 4 2 ϕ 2 + ϕ 4 4 ϕ 2 ϕ 4 4 ϕ 6 7 x Sice the fuctio ϕ 4 x has egative derivative for x. O the other had, we have the iequality log ϕ > 4. Therefore ϕ 6 equatios 6 ad 7 give A + A > 0 if. The theorem is proved. Theorem 2.7 The followig asymptotic formulae ad its hold. i= log F i = 2 2 log ϕ + ϕ 2 log + C + o, 8

6 264 Reza Farhadia ad Rafael Jakimczuk where C = log + k+. 9 k= where i= F i C ϕ 2 2 ϕ 2, 0 C = + k+. k= F F 2 F 2 = ϕ. 2 F F 2 F F = 0. 3 F F 2 F F = ϕ. 4 Let k be a positive iteger, we have log F3 log F 4 log F log F = e log F3 3k log F 4 4k log F k k+ k+ log F = e k+ 6 log F+ = e log F Proof. We have Biet s formula F = ϕ + + = ϕ + + ϕ ϕ 2 Therefore F ϕ, 7

7 Results for the Fiboacci sequece usig Biet s formula 26 ad Hece i= log F = log ϕ 2 log + log ϕ 2 log F i = 2 2 log ϕ + ϕ 2 log + log k= + k+ Note that the series log + k+ = fk k+ k= k= + o. where fk by Lemma 2. coverges absolutely. This proves equatios 8 ad 9. Equatios 0 ad are a immediate cosequece of equatios 8 ad 9. Limit 2 is a immediate cosequece of equatio 0. Limits 3 ad 4 are a immediate cosequece of equatios 0 ad 6. Equatio 7 gives From the Stirlig s formula! log log F = log + log log ϕ + o 9 2π e we obtai log i = log + o 20 i= Therefore see 8 ad 9 log F3 log F 4 log F log = log F log log F log log F log log F = log i + log log ϕ + o i= log + log log ϕ + o = + o That is, equatio. The proof of equatio 6 is the same as the proof of equatio. Note that we have the well-kow asymptotic formula k + 2 k + + k k+ k + ad sice the fuctio x k log x is strictly icreasig ad itegratio by parts we have i k log i = x k log x dx+o k log = k+ log i= k + k + 2 k+ +o k+

8 266 Reza Farhadia ad Rafael Jakimczuk The theorem is proved. Ackowledgemets. Rafael Jakimczuk is very grateful to Uiversidad Nacioal de Lujá. Refereces [] N. N. Vorobèv, Fiboacci Numbers, Spriger, Birkhäuser Verlag, Basel, Received: April 9, 208; Published: April 24, 208