Fibonacci Identities as Binomial Sums

Transcription

1 It. J. Cotemp. Math. Sceces, Vol. 7, 1, o. 38, Fboacc Idettes as Bomal Sums Mohammad K. Azara Departmet of Mathematcs, Uversty of Evasvlle 18 Lcol Aveue, Evasvlle, IN 477, USA E-mal: Abstract To facltate rapd umercal calculatos of dettes pertag to Fboacc umbers, we preset each detty as a bomal sum. Mathematcs Subject Classfcato: 5A1,11B39 Keywords: Fboacc umbers, Fboacc dettes, Fboacc sequece, Pascal s detty, Pascal s Khayyām-Pascal s) tragle 1. Prelmares The most promet lear homogeeous recurrece relato of order two wth costat coeffcets s the oe that defes Fboacc umbers or Fboacc sequece). It s defed recursvely as F + = F +1 + F, where F =,F 1 =1, ad.

2 187 M. K. Azara It s well-kow that the fucto gx) = x 1 x x 1) geerates Fboacc sequece. Bckell ad Hoggatt [] stated that 1) ca be verfed by log dvso. But, sce the method of log dvso s a log process, especally for a large, the author [1] used the method of geeratg fuctos to verfy 1) whch s qucker, regardless of the value of, ad obtaed F = 1 5 [ 1+ 5 ) 1 ] 5 ),. ) Fdg the exact value of F from ) requres multple steps of busy ad messy algebrac calculatos whch s ot desrable. So, our goal ths ote s to preset F as a bomal sum for quck umercal calculatos. Lkewse, we use ths bomal sum to wrte some well-kow ad fudametal dettes cocerg Fboacc umbers as bomal sums as well. It s kow that Fboacc umbers are the sum of the umbers alog the rsg dagoals of Pascal s Khayyam-Pascal s) tragle, ad f we wrte the elemets of Pascal s tragle as bomal terms we have ) S 1 = =1 ) 1 S = =1 ) ) 1 S 3 = + = ) 1 ) 3 S 4 = + =3 ) 1 ) ) 4 3 S 5 = + + =5 ) 1 ) ) S 6 = + + =8 1

3 Fboacc dettes as bomal sums ) ) ) 1 S +1 = ,. 1 ) ) ) +1 Now, usg Pascal s detty = +, r ) r r r 1 we ca easly verfy that S + = S +1 + S, ad hece the bomal sum S satsfes the Fboacc relato. I practce, we eed to kow all terms the bomal sum of S. By specto we ca see that for, S +1 = ) + ) ) ) + where represets the floor fucto. Aga, f we use Pascal s detty, we ca easly show that the above S does deed satsfy the Fboacc relato. ) = ),. Idettes It s well-kow that the left-had sde of each detty Corollary 1 ca be wrtte as a power of a ) sgle Fboacc umber. For example, as early as 1876 Lucas has show that 1 + F = F +, 1+ F = F +1, ad =1 =1 F 1 = F. Oe could use the prcple of mathematcal ducto, =1 combatoral argumet, or just smple algebra to verfy the valdty of these dettes. These Fboacc dettes have bee developed over the cetures by mathematcas ad umber ethusasts alke, ad ther proofs ca be foud varous sources. So, as we stated earler, the goal of ths ote s to wrte some of these fudametal dettes as bomal sums for quck umercal calculatos.

4 1874 M. K. Azara Theorem 1. If F s ay Fboacc umber, the ) ) ) 1 F +1 = ) ) ) =,. Proof follows from our above dscusso. Also, Theorem 1 ca be prove by usg the prcple of mathematcal ducto or combatoral methods. As a drect cosequece of Theorem 1 ad the defto of Fboacc umbers we obta the followg corollary. Corollary 1. If s ay oegatve teger, the +1 ) +1 ) 1+ F = ) ) 1+ F = +1 ) +1 ) F +1 = 3+3 ) 3 +3 v) 1+ F 3+ = 3+ ) 3 + v) F 3+1 = 3+1 ) 3 +1 v) 1+ F 3 = [ ) ] v) 1+ F 4 = +1 ) ) +1 v) F +1 = +1 ) + x) 1+F +1 F + + F F +1 =

5 Fboacc dettes as bomal sums ) +3 x) + +1)F + F = x) F F+3 F ) + 3 = 1)+1 +1 ) +1 x) F +3 F F+1 3 = 1) Cocluso We preseted just a few wdely-kow Fboacc dettes as bomal sums. We are hopg that ths artcle would motvate the curous reader to wrte her/hs favorte Fboacc dettes as bomal sums too. The author hmself wll be workg o wrtg some other Fboacc ad Lucas dettes as bomal sums as well. Refereces [1] Mohammad K. Azara, The Geeratg Fucto for the Fboacc Sequece, Mssour Joural of Mathematcal Sceces, Vol., No., Sprg 199, pp [] Marjore Bckel ad Verer E. Hoggatt, Fboacc s Problem Book, The Fboacc Assocato, [3] Verer E. Hoggatt ad Gerald E. Bergum, A Problem of Fermat ad the Fboacc Sequece, The Fboacc Quarterly, Vol. 15, No. 4, Oct. 1977, pp [4] K. Subba Rao, Some Propertes of Fboacc Numbers, The Amerca Mathematcal Mothly, Vol. 6, No. 1, Dec. 1953, pp [5] Nel. J. Sloa,

6 1876 M. K. Azara Receved: Aprl, 1