Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
MAYFEB Journal of Mathematics Vol 3 (2016) - Pages 10-15 Some Identities for Even and Odd Fibonacci and Lucas Numbers Yogesh Kumar Gupta School of Studies in Mathematics, Vikram University, Ujjain (M. P.), India [email protected] Mamta Singh Department of Mathematical Science and Computer Applications, Bundelkhand University, Jhansi (U.P) [email protected] Omprakash Sikhwal Devanshi Tutorial, Keshw Kunj, Mandsaur (M.P.), India [email protected] Abstract- The Fibonacci sequence are well known and widely investigated. The Fibonacci and Lucas sequences have enjoyed a rich history. To this day, interest remains in the relation of such sequences to many fields. In this paper, we obtain some new properties for Fibonacci and Lucas numbers in terms of even and odd numbers, using Binet’s formula. Keywords- Fibonacci numbers, Lucas numbers, Binet’s formula. I. INTRODUCTION Mathematics can be considered as the underlying order of the universe, and the Fibonacci numbers is one of the most fascinating discoveries made in the mathematical world. Among numerical sequences, the Fibonacci sequence has achieved a kind of celebrity status and has been studied extensively in number theory, applied mathematics, physics, computer science, and biology [2]. The Fibonacci numbers are famous for possessing wonderful and amazing properties. A similar interpretation also exists for Lucas sequence. The Fibonacci numbers have been studied both for their applications and the mathematical beauty of rich and interesting identities that they satisfy. The Fibonacci sequence Fn [4] is defined by the recurrence relation Fn1 Fn Fn1, where n 1 with the initial conditions F0 0 and F1 1 . The Lucas sequence Ln , [7] considered as a companion to Fibonacci sequence, is defined recursively by Ln 1 Ln Ln 1 , where n 1 with the initial conditions L0 2 and L1 1 . n It is well known that F n 1n 1 Fn and L n 1 Ln , for every n ߳চ. The Binet’s formula [6] for Fibonacci numbers can be written in the general form as% 1 5 1 5 n Fn n 2n 5 where n 1. (1.1) The Binet’s formula [6] for Lucas numbers can be written in the general form as% n 1 5 1 5 Fn 2 2 n where n 1. 10 (1.2) MAYFEB Journal of Mathematics Vol 3 (2016) - Pages 10-15 M. Thongmoon [1], [2] has given the identities for the even and odd Fibonacci and Lucas numbers, he also given the common factors of Fibonacci and Lucas numbers. Y.T. Ulutas and N. Omur [8] gave identities F2n and F2n 1 . In this paper, we also obtained some new properties for Fibonacci and Lucas numbers in terms of even and odd numbers. II. PROPERTIES FOR FIBONACCI NUMBERS Theorem (2.1) For every n≥1, 1 {FF2n1 11,, Fn1Ln F 2 n1 2n 1 n n even n odd Proof. By Binet's formula (1.1) and (1.2), we have 1 Fn 1 L n 1 1 5 n 1 1 5 n 1 1 2 n 1 5 5 n 1 5 2 n n 1 1 5 n 1 1 5 n 1 5 n 2 n 1 5 5 5 2 2n 2 n 1 1 5 2 n 1 1 5 1 5 2 2 n 1 5 22 n 1 5 1 2 5 5 F 2 n 1 1 n . Fn 1 Ln { F2n 1 1 , n even F2n 1 1, n odd Theorem (2.2) For every n≥1, Fn 1 Ln F2n 1 1n { F2n 1 1 , n even F2n 1 1, n odd Proof. n n n 1 n 1 1 5 1 5 1 5 1 5 Fn 1 L n 2 2 2 n 1 5 n 1 n 1 n n 1 5 1 5 1 5 1 5 2n 2 n 1 5 2 n 1 2 n 1 1 5 1 5 1 5 1 5 2 n 1 2 5 22 F2 n 1 1n . Fn 1Ln F2n 1 1, n even F2n 1 1, n odd { 11 n 21 5 21 5 4 5 MAYFEB Journal of Mathematics Vol 3 (2016) - Pages 10-15 Theorem (2.3) For every n≥1, Fn 1Ln 2 F2n 3 1n 1 F2n 3 1, n even F2n 3 1, n odd { Proof. By Binet's formula (1.1) and (1.2), we have 1 Fn 1 L n 2 5 2 n 1 5 5 5 1 n 1 1 5 n 1 1 n2 1 5 2 n2 n 1 1 5 n 1 1 5 n 2 1 5 n 2 2 n 1 5 1 5 2 2n2 2 n 3 1 5 2 n 3 1 5 1 5 2 2n 3 5 22 n 1 1 5 1 2 5 5 F 2 n 3 1n 1 . Fn 1Ln 2 F2 n3 1, n even F2 n3 1, n odd { Theorem (2.4) For every n≥1, Fn 2 Ln 1 F2n 3 1n 1 F2n3 1, {F 2 n 3 n even 1, n odd Proof. 1 Fn 2 L n 1 1 1 5 n 2 1 5 n 2 1 2n2 5 n 2 1 5 n 2 1 2n2 5 5 2 n 3 1 5 2 n 3 2 2n 3 5 n 1 5 n 1 n 1 n 1 5 1 5 2 n 1 n 1 21 1 5 1 5 22 5 2 F 2 n 3 1 n 1 . F2n3 1, {F Fn 2 Ln 1 2 n 3 n even 1, n odd Theorem (2.5) For every n≥1, Fn 3 Ln F2n3 2, {F 2 n 3 2, 1 5 2 n even n odd 12 5 21 4 5 5 MAYFEB Journal of Mathematics Vol 3 (2016) - Pages 10-15 Proof. 1 Fn 3 L n 1 5 n 3 1 2n3 5 5 5 n 3 1 2n3 1 n 3 1 5 5 n 3 1 5 5 2 n 3 1 2 2n 3 5 2 5 n 5 n 2n 5 1 5 2 n 1 2 n 3 1 5 n 1 5 22 n 16 5 8 5 F 2 n 3 2 1 n . Fn 3 Ln F2n3 2, {F 2 n 3 n even 2, n odd Theorem (2.6) For every n≥1, Fn 1Ln 3 F2n 4 1n 1 F 2 F2 n4 F2 , n even {F 2 n 4 F 2 , n odd Proof. By Binet's formula (1.1) and (1.2), we have 1 Fn 1 L n 3 1 1 5 n 1 1 5 2 n 1 5 5 n 1 1 2 n 1 5 5 n 1 1 n 1 1 2 n 4 1 2 2n 4 5 5 5 5 2 1 5 2 n 3 1 5 2n3 2 n 4 1 5 n3 5 1 5 22 n3 n 3 n 1 1 F 2 n 4 1 n 1 F 2 . Fn 1Ln 3 F2n4 F2 , n even F2n4 F2 , n odd { III. PROPERTIES FOR LUCAS NUMBERS Theorem (3.1) For every n≥1, L2 n 2, {L 5Fn2 L2n 2 1n 2n 2, n even n odd 13 5 2 1 22 5 5 2 MAYFEB Journal of Mathematics Vol 3 (2016) - Pages 10-15 Proof. By Binet's formula (1.1) and (1.2), we have 2 n n 1 5 1 5 2 5 Fn 5 2n 5 n n n 2 2n 1 5 1 5 2 1 5 1 5 2 2n 2 2n n 1 5 1 5 L2n 2 22 n L 2 n 2( 1) . 5Fn2 L2n 2, {L 2n n even 2, n odd Theorem (3.2) For every n≥1, 5Fn1 Fn L2 n1 1 {LL22 nn11 11,, n n even n odd Proof. (1 5 ) n 1 (1 5 ) n 1 (1 5 ) n (1 5 ) n 5 Fn 1 Fn 5 2 n 1 5 2n 5 (1 5 ) 2 n 1 (1 5 ) 2 n 1 (1 5 ) n 1 (1 5 ) n 2 2 n 1 2 2 n 1 (1 5 ) n (1 5 ) n (1 5 ) (1 5 ) L 2 n 1 2 2 2 2n L 2 n 1 ( 1) n . 5Fn1 Fn {LL22 nn11 11,, n even n odd Theorem (3.3) For every n≥1, 5Fn1 Fn L2 n1 1 {LL22 nn11 11,, n n even n odd Theorem (3.4) For every n≥1, 5 Fn2 1 L 2 n 2 2 1 n 1 Theorem (3.4) For every n≥1, n even n odd L2n2 2, n even L2n2 2, n odd { 5F 2 L2n 2 2 1n 1 n 1 { LL 22 nn 22 22 ,, Similar proof can be given for remaining parts (3.3) to (3.4). 14 (1 5 ) n 1 (1 5 ) n 2 2 n 1 MAYFEB Journal of Mathematics Vol 3 (2016) - Pages 10-15 IV. CONCLUSION This paper describes identities for Fibonacci and Lucas numbers by their Binet’s formula. These identities can be used to develop new identities of Fibonacci and Lucas numbers in terms of even and odd numbers through Binet’s formula. ACKNOWLEDGMENT The authors are grateful to the referees for their useful comments. REFERENCES [1] [2] [3] [4] [5] [6] [7] M. Thongmoon, Identities for the Common factors of Fibonacci and Lucas numbers, International Mathematical Form, Vol. 4, No. 7 (2009), 303308. M. Thongmoon, Identities for the Even and Odd Fibonacci and Lucas numbers, Int. J. Contemp. Math. Sciences, Vol. 4, No.14 (2009), 671-676. M. Singh, Y.K. Gupta, O. Sikhwal, Generalized Fibonacci-Lucas Sequences its Properties, Global Journal of Mathematical Analysis, 2(3), 2014, 160-168. M. Singh, Y.K. Gupta, and K. Sisodiya, Generalization of Fibonacci Sequence and Related Properties, “Research Journal of Computation and Mathematics, Vol. 3, No. 2, (2015), 12-18. M. Singh, Y.K. Gupta, O. Sikhwal, “Identities of Generalized Fibonacci-Like Sequence.” Turkish Journal of Analysis and Number Theory, Vol. 2, No. 5 (2014): 170-175. doi:10.12691/tjant-2-5-3. S. Vajda, Fibonacci and Lucas numbers and the golden section, Ellis Horwood Limited Publ., England, (1989). T. Koshy, Fibonacci and Lucas numbers with applications, John Wiley and Sons, Proc., New York-Toronto, (2001). F F Y.T. Ulutas, and N. Omur, New Identities for 2 n and 2 n 1 , International Forum, Vol. 3, No.3 (2008), 147-149. Y.K. Gupta, M. Singh, O. Sikhwal, Generalized Fibonacci – Like Sequence Associated with Fibonacci and Lucas Sequences, Turkish Journal of Analysis and Number Theory, Vol. 2, No. 6 (2014), 233-238. [10] Y.K. Gupta, V. H. Badshah ,M. Singh, K. Sisodiya, Generalized Additive Coupled Fibonacci Sequences of Third order and Some Identities’’, International Journal of Innovative Science, Engineering & Technology, Vol. 2 Issue 3, March 2015, 80-55. [11] Y.K. Gupta, O. Sikhwal, and M. Singh, Determinantal Identities of Fibonacci, Lucas and Generalized Fibonacci-Lucas Sequence, MAYFEB Journal of Mathematics, Vol. 2 (2016) - Pages 17-23. [12] Y.K. Gupta, O. Sikhwal, and K. Sisodiya, Determinantal Identities of Generalized Fibonacci-Like Sequence, MAYFEB Journal of Computer Science, Vol. 1 (2017) - Pages 1-6. [8] [9] 15