Download Some Identities for Even and Odd Fibonacci and Lucas Numbers

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
Fn1  Fn  Fn1, 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   1n 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  {FF2n1 11,,
Fn1Ln  F
2 n1
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   1n 
{
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   1n .
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   1n 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 
n2
1 5 


 2 


n2 



n 1  1  5 n 1   1  5 n  2  1  5 n  2 


2 n 1 5

1



5 
2 
2n2
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   1n  1 .
Fn 1Ln  2 
F2 n3 1,
n even
F2 n3 1,
n odd
{
Theorem (2.4) For every n≥1,
Fn  2 Ln 1  F2n  3   1n 1 
F2n3 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 
2n2
5
n  2  1  5 n  2   1 
2n2
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 .
F2n3 1,
{F
Fn  2 Ln 1 
2 n 3
n even
1,
n odd
Theorem (2.5) For every n≥1,
Fn  3 Ln 
F2n3  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 
2n3
5

5
 
  
5
n  3  1 
2n3

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 
F2n3  2,
{F
2 n 3
n even
 2,
n odd
Theorem (2.6) For every n≥1,
Fn 1Ln  3  F2n  4   1n 1 F 2 
F2 n4  F2 ,
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
2n3
2 n  4    1 
5
n3


5
 1 
5
22
n3 



n  3 







n 1


 1


 F 2 n  4   1 n  1 F  2 .
Fn 1Ln  3 
F2n4  F2 ,
n even
F2n4  F2 ,
n odd
{
III. PROPERTIES FOR LUCAS NUMBERS
Theorem (3.1) For every n≥1,
L2 n  2,
{L
5Fn2  L2n  2 1n 
2n
 2,
n even
n odd
13
5
 2  1 
22
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,
5Fn1 Fn  L2 n1   1  {LL22 nn11 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 .
5Fn1 Fn  {LL22 nn11 11,,
n even
n odd
Theorem (3.3) For every n≥1,
5Fn1 Fn  L2 n1   1  {LL22 nn11 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
L2n2  2,
n even
L2n2  2,
n odd
{
5F 2  L2n  2  2 1n 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]
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