research article from fibonacci sequence to the golden...

Hindawi Publishing CorporationJournal of MathematicsVolume 2013, Article ID 204674, 3 pageshttp://dx.doi.org/10.1155/2013/204674

Research ArticleFrom Fibonacci Sequence to the Golden Ratio

Alberto Fiorenza1,2 and Giovanni Vincenzi3

1 Dipartimento di Architettura, Universita di Napoli, Via Monteoliveto 3, 80134 Napoli, Italy2 Istituto per le Applicazioni del Calcolo “Mauro Picone”, Sezione di Napoli, Consiglio Nazionale delle Ricerche,Via Pietro Castellino 111, 80131 Napoli, Italy

3 Dipartimento di Matematica, Universita di Salerno, Via Ponte Don Melillo 4, Fisciano, 84084 Salerno, Italy

Correspondence should be addressed to Alberto Fiorenza; [email protected]

Received 14 November 2012; Accepted 11 February 2013

Academic Editor: Mike Tsionas

Copyright © 2013 A. Fiorenza and G. Vincenzi.This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

We consider thewell-known characterization of theGolden ratio as limit of the ratio of consecutive terms of the Fibonacci sequence,and we give an explanation of this property in the framework of the Difference Equations Theory. We show that the Golden ratiocoincides with this limit not because it is the root with maximum modulus and multiplicity of the characteristic polynomial, but,from a more general point of view, because it is the root with maximum modulus and multiplicity of a restricted set of roots,which in this special case coincides with the two roots of the characteristic polynomial. This new perspective is the heart of thecharacterization of the limit of ratio of consecutive terms of all linear homogeneous recurrences with constant coefficients, withoutany assumption on the roots of the characteristic polynomial, which may be, in particular, also complex and not real.

In this paper, we consider a well-known property of theFibonacci sequence, defined by

𝐹0= 𝐹1= 1, 𝐹

𝑛= 𝐹𝑛−1+ 𝐹𝑛−2, 𝑛 > 1, (1)

namely, the fact that the limit of the ratio of consecutive terms(the sequence defined from the ratio between each term andits previous one) is Φ, the highly celebrated Golden ratio:

lim𝑛→∞

𝐹𝑛+1

𝐹𝑛

= Φ. (2)

Many proofs already exist and arewell known since long time,and we do not wish to add one more to the repertory.

The Fibonacci sequence can be studied in the frameworkof the Difference Equations Theory (e.g., see [1] Chapter 3Page 43), where, roughly speaking, the properties of themoregeneral sequences (𝐹

𝑛) satisfying

𝐹𝑛+ 𝑎𝑘−1𝐹𝑛−1+ ⋅ ⋅ ⋅ + 𝑎

0𝐹𝑛−𝑘= 0 ∀𝑛 ≥ 𝑘, (3)

where 𝑎0= 0, 𝑎1, . . . , 𝑎

𝑘−1∈ C, are studied. The first step of

the theory consists in considering the associated characteris-tic polynomial

𝑝 (𝜆) = 𝜆𝑘+ 𝑎𝑘−1𝜆𝑘−1+ ⋅ ⋅ ⋅ + 𝑎

0, (4)

whose complex roots 𝜆1, . . . , 𝜆

ℎ, with respective multiplicity

𝑘1, . . . , 𝑘

ℎ, 𝑘1+ ⋅ ⋅ ⋅ + 𝑘

ℎ= 𝑘 permit to express explicitly the

terms 𝐹𝑛by means of the representation, known as Binet’s

formula:

𝐹𝑛= 𝑐1,1𝜆𝑛

1+ 𝑐1,2𝑛𝜆𝑛

1+ ⋅ ⋅ ⋅ + 𝑐

1,𝑘1

𝑛𝑘1−1𝜆𝑛

1+ ⋅ ⋅ ⋅ + 𝑐

ℎ,1𝜆𝑛

ℎ

+ 𝑐ℎ,2𝑛𝜆𝑛

ℎ+ ⋅ ⋅ ⋅ + 𝑐

ℎ,𝑘ℎ

𝑛𝑘ℎ−1𝜆𝑛

ℎ,

(5)

where 𝑐𝑖,𝑗

are (complex) numbers uniquely determined by𝐹0, . . . , 𝐹

𝑘−1. In the case of the Fibonacci sequence, (3), (4),

and (5) become, respectively,

𝐹𝑛− 𝐹𝑛−1− 𝐹𝑛−2= 0, (6)

𝑝 (𝜆) = 𝜆2− 𝜆 − 1, (7)

𝐹𝑛= 𝑐1,1Φ𝑛+ 𝑐2,1(1 − Φ)

𝑛=Φ

√5Φ𝑛−1 − Φ

√5(1 − Φ)

𝑛, (8)

where the last equality comes out taking into account theinitial conditions 𝐹

0= 𝐹1= 1. Well, the fact that the

Golden ratio, limit of the ratio of adjacent terms, coincideswith one of the roots of the characteristic polynomial (7)

2 Journal of Mathematics

is not a coincidence. It can be very easily seen that this isalways the case, also for general order-k linear homogeneousrecurrences with constant coefficients; the limit of the ratio ofadjacent terms (in the following, we will refer to this limit asthe Kepler limit of (𝐹

𝑛)), if it exists, is always one of the roots

of the characteristic polynomial. It suffices to write, startingfrom (3),𝐹𝑛

𝐹𝑛−1

𝐹𝑛−1

𝐹𝑛−2

⋅ ⋅ ⋅𝐹𝑛−𝑘+1

𝐹𝑛−𝑘

+ 𝑎𝑘−1

𝐹𝑛−1

𝐹𝑛−2

⋅ ⋅ ⋅𝐹𝑛−𝑘+1

𝐹𝑛−𝑘

+ ⋅ ⋅ ⋅ + 𝑎0= 0

(9)

and to substitute each ratio with its limit.The question is:

Q: if for an order-k linear homogeneous recurrence withconstant coefficients, the limit of the ratio of adjacentterms exists, which root is it?

In the case of the Fibonacci sequence, it is clear that theGolden ratio is the root of the characteristic polynomial withmaximummodulus, and all the proofs of (2) use more or lessimplicitly this property. And the same machinery works withseveral other examples of recurrences (e.g., see the pioneeringpaper [2]).

Surprisingly, the correct answer to the question is notthat the limit of the ratio of adjacent terms is the root withmaximum modulus. The fact that the Golden ratio is theunique root of maximum modulus is just a coincidence! Infact, consider, for instance, the sequence

𝐹0= 1, 𝐹

1= 1 − Φ,

𝐹𝑛= 𝐹𝑛−1+ 𝐹𝑛−2, 𝑛 > 1,

(10)

with different initial conditions and same characteristicpolynomial (7). Here, Binet’s formula reads 𝐹

𝑛= (1 − Φ)

𝑛

(the term containing Φ𝑛 is in some sense “killed” by theinitial conditions). In this case, the sequence is geometric,and the Kepler limit is the other root of the characteristicpolynomial: the smaller. This sequence (along with thoseobtained multiplying each term by a nonzero constant) isknown as an “exception”, and much attention is devotedin the literature to the roots of maximum modulus; up tosome assumptions which should exclude the exceptions, itis known that the Kepler limit exists if, among the rootsof maximum modulus of (4), there exists a unique rootof maximal multiplicity (the “dominant” root; see [3] andreferences therein).

Another surprise: the existence of the dominant root isindependent of the existence of the Kepler limit, and also thevalue of the limit is not necessarily the dominant root.This isshown by the following examples. Consider the sequence

𝐹0= 3, 𝐹

1= 1,

𝐹2=−1, 𝐹

3= 1,

𝐹𝑛= 3𝐹𝑛−1− 3𝐹𝑛−2+ 3𝐹𝑛−3− 2𝐹𝑛−4, 𝑛 > 3;

(11)

the characteristic polynomial is 𝑝(𝜆) = 𝜆4−3𝜆3+3𝜆2−3𝜆+2,whose roots are 2, 1, −𝑖, 𝑖; Binet’s formula is

𝐹𝑛= 1𝑛+ (−𝑖)

𝑛+ 𝑖𝑛. (12)

In this case, the dominant root exists, while the Kepler limitdoes not. In the case

𝐹0= 1, 𝐹

1= 1, 𝐹

2= 1,

𝐹𝑛= 𝐹𝑛−1− 4𝐹𝑛−2+ 4𝐹𝑛−3, 𝑛 > 2,

(13)

the characteristic polynomial is𝑝(𝜆) = 𝜆3−𝜆2+4𝜆−4, whoseroots are −2𝑖, 2𝑖, 1. Binet’s formula is

𝐹𝑛= 1𝑛. (14)

Here, the characteristic polynomial has 𝑛𝑜𝑡 a dominant root,but there exists the Kepler limit, which is the root 1 (note that1 is not among the roots of maximummodulus).

These phenomena attracted several researchers sincePoincare, who proved in a more general context (e.g., see [4])the following result.

Theorem A. Let 𝜆1, . . . , 𝜆

𝑘be the roots of the characteristic

equation

𝜆𝑘+ 𝑎𝑘−1𝜆𝑘−1+ ⋅ ⋅ ⋅ + 𝑎

0= 0 (15)

of (3), and suppose that |𝜆𝑖| = |𝜆𝑗| for 𝑖 = 𝑗. Then, either 𝐹

𝑛= 0

for all large n or there exists an index 𝑖 ∈ {1, . . . , 𝑘} such that

lim𝑛→∞

𝐹𝑛+1

𝐹𝑛

= 𝜆𝑖. (16)

The condition about the pairwise distinct moduli isoptimal (namely, necessary and sufficient) if one requires theexistence of the Kepler limit for all possible initial conditions;in other words, if there exist two distinct roots with thesame modulus, then it is always possible to consider initialconditions such that the Kepler limit does not exist.

What can be said about the general case, also when themoduli of the roots are not necessarily distinct? Is there anecessary and sufficient condition for the existence of theKepler limit?

Well, for a given linear recurrence with constant coeffi-cients (𝐹

𝑛), consider the “essential Binet’s formula represen-

tation”; namely, write it in the form (5) where the coefficients𝑐𝑖,𝑗have been computed starting from the initial conditions,

and where only the nonzero coefficients 𝑐𝑖,𝑗appear:

𝐹𝑛= ∑

𝑙

𝑐𝑖𝑙,𝑗𝑙

𝑛𝑗𝑙−1𝜆𝑛

𝑖𝑙

. (17)

In this representation, one can see only a 𝑠𝑢𝑏𝑠𝑒𝑡 of the 𝑘addends of the right hand side of (5). In some sense, theaddends not appearing in the right hand side of (17) canbe considered as “killed” by the initial conditions. In thisrepresentation, among the addends containing the “survived”roots, count how many, among them, have maximum mod-ulus and contain the maximum power of 𝑛 (let us call them“leading”). For instance, the classical Fibonacci sequence (1)admits the Golden Ratio Φ as unique leading root (see (8));the sequence (10) has the same characteristic polynomial of(1), but the unique leading root is 1−Φ; in the case of (11), the

Journal of Mathematics 3

leading roots are 1, −𝑖, 𝑖 (see (12)). Moreover, if we considerthe sequence

𝐹0= 1, 𝐹

1= −3, 𝐹

2= −11,

𝐹𝑛= 6𝐹𝑛−1− 11𝐹

𝑛−2+ 6𝐹𝑛−3, 𝑛 > 2,

(18)

the characteristic polynomial is 𝑝(𝜆) = (𝜆 − 3)(𝜆 − 2)(𝜆 − 1),and Binet’s formula is

𝐹𝑛= 5 ⋅ 1

𝑛+ (−4) ⋅ 2

𝑛, (19)

the unique leading root being 2.In [3], it is proved that if you count exactly one leading

term, then the Kepler limit exists, and in such case, the Keplerlimit is exactly the corresponding root (“the unique leadingroot”). When this happens, we say that the initial conditionsare in agreement with the characteristic polynomial.

Thus, Theorem A is, of course, an immediate corollary;it is trivial that in the case of pairwise distinct moduli,independently of the “killed” roots, the set of the “survived”roots, whatever it is, must contain a unique leading root.

On the other hand, for a general order-k linear homoge-neous recurrence with constant coefficients, it may happenthat (e.g., see (11)) there aremore than one leading root.Whatcan be said, in this case, about the existence of the Keplerlimit? A third surprise, in this case, is that the Kepler limitdoes not exist; the condition given earlier (i.e., the uniquenessof the leading root) is also necessary.

In the end, the notion of agreement gives us the answer tothe original question Q: the unique leading root.

The importance of this study relies upon the fact thatthe ratio of consecutive terms tells that all linear recurrencesin agreement behave at infinity like geometric sequences,and the first terms (which give the initial conditions), alongwith the law (which can be conjectured after a reasonablenumber of tests), permit to predict the ratio. This may bevery important for all applications, where linear recurrencesrepresent mathematical models.

References

[1] W. G. Kelley and A. C. Peterson, Difference Equations, Har-court/Academic Press, SanDiego, Calif, USA, 2nd edition, 2001,An introduction with applications.

[2] E. P. Miles, Jr., “Generalized Fibonacci numbers and associatedmatrices,”TheAmericanMathematicalMonthly, vol. 67, pp. 745–752, 1960.

[3] A. Fiorenza and G. Vincenzi, “Limit of ratio of consecutiveterms for general order-𝑘 linear homogeneous recurrences withconstant coefficients,”Chaos, Solitons&Fractals, vol. 44, no. 1–3,pp. 145–152, 2011.

[4] H. Poincare, “Sur les equations lineaires aux differentiellesordinaires et aux differences finies,” American Journal of Math-ematics, vol. 7, no. 3, pp. 203–258, 1885.

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