ON SUMMATIONS AND EXPANSIONS OF FIBONACCI NUMBERS

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1 ON SUMMATIONS AND EXPANSIONS OF FIBONACCI NUMBERS HERTAT. FRESTAG Hoilis, Virgiia INTRODUCTION Oe of the early delights a eophyte i the study of Fiboacci umbers experieces may be a ecouter with some elemetary summatio properties such as il> F i F As soo as his curiosity is aroused, he may wish to ivestigate summatios which "skip" a costat umber of Fiboacci umbers, for istace the problem of obtaiig a formula for the sum of the first Fiboacci umbers of odd positio idex e But as has ofte bee observed mathematicias are like lovers; give them the little figer, ad they will wat the whole had. Ca oe fid a relatioship which spells out the sum of ay fiite Fiboacci sequece whose subidices follow the patter of a arithmetic progressio? A SUMMATION THEOREM (Theorem 1) Seekig a patter for the sum of a umber of equally spaced Fiboacci umbers meas a cocer with y ^ F, (. = ki+r), i=o * r is a o-egative iteger s whereas k is a atural umber. Let us use the Biet formula a 11 - b.., 1 + «st5,, 1 - NT5 F = with a = - 7T ad b = ^. 2 2 </5 We also ote that ab = - 1. The Lucas umber, L, is L = a + b. The 2X X 63

2 64 ON SUMMATIONS AND EXPANSIONS OF FIBONACCI NUMBERS [Feb. becomes: -±-Y (a ik+r -. ik+r x - b ) = - _L 45 ' r a ( + 1 ) k - 1 a - k - a - 1 h r b ( + 1 ) k - 1 b k - l [ a ( + l ) k + r _ ^ k ^ ^ + l ^ r ^ r ^ k ^ Nf5[(-l) k L k ] Performig the idicated operatios ad agai employig the Biet formula, we are ready to give the sum of Fiboacci umbers begiig with F. The sequece cotiues equally spaced such that (k - 1) Fiboacci umbers are left out from ay oe term to the ext* Theorem 1. f F <-!> V l ) k + r + (-l> mi(k ' r) -% r _ k l ~ W + * r t k(i " 1)+r = ("l> k \ where k is ay atural umber ad r ay o-egative iteger. by:, _ \ 0, whe r < k I 1, whe r > k The umber t is defied Sice F«,, vaishes for r = k 9 t eed ot be defied i this case. r - k Attetio should be draw to the fact that we may restrict r to the coditio 0 < r < k by the Reductio Formula: (2) the If r = r (mod k), i. e. : r = ak + r where a is a atural umber ad 0 < r < k 9 2L» F (i-l)k+r = 2 ^ F(a+i-l)k+r +a a = 2^ F (i-i)k+r " JLS F (i-l)k+r 9 While the restrictio o r" is useful for reductio purposes, it is ot a ecessary coditio for relatioship (2). Special Cases of Theorem 1. We otice that the result of our summatio ivolves a expressio which combies o more tha four terms. Thus 9 this relatioship would be quite helpful wheever is "fairly l a r g e. " For r = 0, the special case

3 1973] ON SUMMATIONS AND EXPANSIONS OF FIBONACCI NUMBERS 65 (3) y F = ( " 1} F k + F k - F (-H)k M ^ L k tl may merit attetio. It is evidet that Theorem 1 embraces the basic elemetary summatio formulas of this kid 0 Obviously, k = 1, r = 0 yields: which is the formula we previously quoted for the sum of the first Fiboacci umbers. However, it is aesthetically satisfyig that the summatio formulas for the first Fiboacci umbers of odd idices ad those of eve idices also become special cases of our patter. Thus, by lettig k = 2 ad r = 0, we get whereas r = 1 yields: 2-J F 22i i ~ F "2+l 2-"«- X) F 2i-1 = F 2 ' 1 If oe relatioship combiig the two cases were required, Theorem 1 for k = 2 ad r = 0 or 1 becomes: or, more simply: 2~J F 2(i-l)+r < 4 > X F 2(i-1)H )+r = F " ^ F 2 + r - l F 2 - r " F r + r ~ 1 ' It may be istructive to check other cases of small "skippig umbers' 1 k. Owig to reductio formula (2), the coditio r < k does ot limit the geerality of the results. For k = 3 we obtai 2 F Q _ 1 - ( - l ) r F 0 - F 2L F 3(i-1)H 3+r-l ' 3-r r. )+r

4 66 ON SUMMATIONS AND EXPANSIONS OF FIBONACCI NUMBERS [Feb. which may also be stated as (5) ad, for k = 4, we have Z F 3(i-1) +r F 3 + r-l " i r "! or, alteratively: (6) <L*t A 4 ( i - l ) + r i = l E F 4(i-l) + r = 2 F 4 + r F 4+r-2 " ^ ^ - r r _ 2 J ' _ F r These equivaleces, relatioships (5) ad (6), may easily be verified by straight substitutio of the few r-values to which we are restricted. All of these formulas ca, however, readily be established either by usig the Biet formula, or else, employig mathematical iductio. They were stated here merely as a matter of iterest sice oe of them seem too obvious. Two further observatios may be metioed. We might wish to impose the coditio r = k o Theorem 1. The Ct\ V F - ( " X) F k - F (+l)k + F k Clearly, the summatio formula for the first Fiboacci umbers of eve subidex is a special case of this. It may also be of iterest to ote that o the basis of Theorem 1, L, divides ito all sums of our kid, provided k is odd, i. e., the umber of Fiboacci umbers "skipped over" i our summatio is eve. If this umber were odd, (2 - L, ) would be a divisor of our sum. AN EXPANSION THEOREM (Theorem 2) But has f t Jacobi advised us: "Ma muss immer umkehre" (oe must always tur aroud)? Thus havig obtaied summatio results as expressios ivolvig Fiboacci umbers we may ow experimet with a iversio ad pose the problem: Ca a Fiboacci umber be expaded ito a series remiiscet of a expasio for the power of a biomial? Partly aalogous to Theorem 1, ad primarily for the sake of developig some otios, we symbolize our Fiboacci umbers F as F,, where all letters represet o-egative itegers. The proposed expasio reads:

5 1973] ON SUMMATIONS AND EXPANSIONS OF FIBONACCI NUMBERS 67 Theorem 2. k-1 Z - r l i I m m r + l F m-f-r-k+i-hl> > / ( = km + r) I our proof j we use mathematical iductio o. Symbolizig Theorem 2 by R(), we readily verify R() for the first few atural umbers. Now we eed to show that the correctess of R(s - 1) ad of R(s) implies correctess of R(s + 1), where s represets ay atural umber. This meas that we ivestigate whether equals / k - 1 \ k - i - i i T F + F 1 I i I m ' m+l L m+r-k+i m+r-k+i+l J / k - A F k - ' i - ' i F J. i I i I m m+l i F m+r-k+i+2 However, the iterative defiitio of Fiboacci umbers assures the correctess of this equality ad, hece, completes the proof. assert that As a illustratio, we might wish to expad F which is easily verified. Special Cases of Theorem 2. '--zm'f'i'w X f Some special cases might be poited to. subidex, Theorem 2 reduces to: by lettig m = 3 ad r = 2. We Cosiderig Fiboacci umbers with eve (8» - 1 2» = (^) 2 ' : F - - M 2 J 2 Vi /2) + i But those of odd subidex may be expaded o the basis of (9) F. A Corollary of Theorem / / - 3 \ = S\ t P F Li (9-)/2+i We propose a corollary of expasio formula 2 (Theorem 2) which gives a prescribed umber of terms for the expasio. Let the symbol a stad for that umber. I our

6 68 ON SUMMATIONS AND EXPANSIONS OF FIBONACCI NUMBERS [Feb. coditio = km + r we stipulate that m = 1 ad k = a, ad we obtai: Corollary of Theorem 2. a-1 (10) F M = " " zl* 1 i l F +2(l-a)H )+i ' ^ / where 2 < a < ^ i. Special Cases of the Corollary; The followig two special cases seem worth metioig. We desire to let a be the largest possible umber. Case 1: If is eve, a = /2 is chose. The 2 x -sp; 1 ) X / (ID F M = >, ( 2 7 ^ J F. + 2 ad there are /2 terms i the expasio. Case 2: T J J + 1 If is odd, a = -, -1 2 (12) F = > A 2 }F. +1 V i / ad the expasio has terms. To illustrate, let us expad F 2 i ito a five-term series. The = 21. Usig relatioship (10) ad lettig a = 5, we have: x / 13+i J which is correct. For the maximum umber of terms i the expasio we would desigate a as beig 11 ad use (12). The 10 fiou F 21 = >. I \ 1 F! ( " ) i+1 ' a relatioship which ca also be easily verified.

7 1973] ON SUMMATIONS AND EXPANSIONS OF FIBONACCI NUMBERS 69 BACK TO ANOTHER SUMMATION THEOREM (Theorem 3) Oce agai, we might "ivert. ff Our summatio theorem (Theorem 1) gave us a expasio ivolvig Fiboacci umbers as the result of the additio. Now let us give a summatio which results i oe Fiboacci umber e the best advatage. This problem may possibly use Theorem 2 to Startig with a summatio ivolvig Fiboacci umbers of prescribed idices, ca we predict the resultig Fiboacci umber? Agai recallig Jacobi f s advice, we reverse the expasio of a give Fiboacci umber to a sum. Now desigate a sum which leads to a p r e - dictable Fiboacci umber. Symbolize m by u, ad u + r - ( - r)/u + 1 by v. The r == v - l - u + k ad Theorem 2 becomes: Theorem 3. k-1 E / k - l \ k - l - i i I i J u u+1 v+i (k-1) (u+1) +v for ay arbitrarily chose atural umbers u ad v. The umber k may be ay iteger greater tha or equal to 2. To illustrate this summatio idea, we try a summatio ivolvig F 4 ad F T. Here we let u = 4, ad v = 7, ad get: sfvy- 1-1 * 1 * 7+i " \ ' We predict F^.,? as our result which is correct. Pre-assigig the Fiboacci Number Resultig from Summatio Theorem 3; Formula 3 is a method for a summatio which uses prescribed Fiboacci umbers ad predicts a Fiboacci umber as the result. What about assigig the resultig Fiboacci umber without prescribig Fiboacci umbers ivolved i the summatio? This summatio, ot ecessarily uique, ca be had by cosiderig two cases. Case 1. The Fiboacci umber to be attaied has odd subidex. We choose u = v = 1, ad have k-1 s( k ; 1 < 13 > > A 7 ) h F i + 2 = * *! Case 2. We wish to obtai a Fiboacci umber of eve subidex. let u ad v take o the values 1 ad 2, respectively* Here: For this purpose we

8 70 ON SUMMATIONS AND EXPANSIONS OF FIBONACCI NUMBERS [Feb. k-2 < 14 > 2,1 i Fi,2 = F 2k- 1=0 X / Obviously, the umber of t e r m s i these summatios will be ( + l)/ 2 for odd subidices, ad /2 for eve oes. We realize, however, that our choices for u ad v have forfeited the ability to discer the powers of F ad F which characterize the terms of Theorem 3. Pre-Assigig the Fiboacci Number Resultig from Summatio Theorem 3 as well as the Number of Terms i the Summatio, ad Retaiig Geerality. Fially, we prescribe the resultig Fiboacci umber F as well as k, the umber of terms i the summatio. Moreover, to avoid the difficulty ecoutered above, exclude the somewhat trivial cases which ivolve F A = F 2 = 1 amog the summatio terms. We impose the coditio: u,v > 3. Furthermore, the iterative defiitio of Fiboacci umbers: 2-J *+i +2 iheretly provides a summatio of two terms resultig i a Fiboacci umber (eve though the summatio is ot of our geeral type). Therefore, impose the coditio: k > 3. The, for all > 4k - 1; i. e., for all > 11, we ca do justice to our data by assigig appropriate values to u ad. v such that (15) = (k - l)(u + 1) + v is satisfied. Agai, o claim is made for uiqueess. For example, to obtai F 1A through a summatio of three t e r m s, the followig choice proves successful: X ' 4 F 3+i F l l For a summatio of three terms for F 1 5, we ca already write: ^ ' ^ ' F 2+i = F r 15

9 1973 ON SUMMATIONS AND EXPANSIONS OF FIBONACCI NUMBERS 71 Lack of Uiqueess Predictig the Number of Differet Summatios Ca you foretell the umber of differet summatio represetatios of our type, each havig k t e r m s, ad leadig to the same Fiboacci umber F? Usig relatioship (15), our predictio becomes: If set T is defied by T = t : 4 < t < - ~ 3 k - 1 the the umerosity of T, that i s, the umber m\- predicts the possible umber of differet summatios of our type 5 each havig k terms ad (16) leadig to the Fiboacci umber F. To illustrate, there will be 52 te-term summatios of our kid leadig to F 500. We would have: X ' X 7 X ' 53 F 23+i X ; 4 * 464+i 500 V = F [Cotiued from page 62. ] the thei V = L, the Lucas sequece, ad so (HI) ow gives the correct expressio for (9) i (*). Case 2. A + B = 0. We ow obtai from () (IV) f(x + c t ) - f(x + c 2 ) _ U ci - c 2 =0 X TT ^ where U 0 = 0, U, = 1, ad U + 2 = P U QU^ Thus for P = 1, Q = - 1, U = F ; ad for P = 2, Q = - 1, U = P, the Pell sequece. For m = 1, 2,, we obtai from (IV) (V) f(x + c f ) - f ( x + cf) V m ^r^ = D f(x) =0 Remarks. The same ideas i (*) show that the geeratig fuctio of the momets of the iverse operator [Cotiued o page 84. ]