ELEMENTARY RESULTS ON THE FIBONACCI NUMBERS ROGÉRIO THEODORO DE BRITO Contents 1. Introduction 1 2. Denition and Elementary Properties 1 3. Other Properties of the Golden Ratio 6 4. Algorithms for the Fibonacci Numbers 7 5. “Appearances” of the Fibonacci Numbers 7 6. The Fibonacci numbers and their siblings 9 7. Fibonacci Numbers and Linear Algebra 9 8. Fibonacci Numbers and Their use in Analysis of Algorithms 9 References 12 1. Introduction The Fibonacci Numbers arise in plenty of situations of our days (even if in disguise) and they fascinate a lot of people for the abundance of their properties. It is remarkable that such a sequence presents so many connections with a number of branches of Mathematics, ranging from Complex Variables to Abstract Algebra, from Discrete Mathematics [4] to Linear Algebra, from Elementary Number Theory [1] to Analysis of Algorithms [2], just to cite a few. It is the purpose of these short notes to present (in a very easy way—perhaps, accessible even to high-school students) some of the most widely known and elementary facts regarding such intriguing numbers. The reader can nd many other sources of material for further study (including a journal entirely devoted to Fibonacci Numbers, the Fibonacci Quarterly Journal!) in the references. They contain a number of much deeper results that are outside the scope of these modest notes. 2. Definition and Elementary Properties The Fibonacci Numbers can be dened in many dierent (and equivalent) ways, but the most common denition seemsDRAFT to be the fact that they are a sequence of integers that can be dened by a very simple recurrence relation: 8 > 0; if i = 0 <> Fi = 1; if i = 1 > : Fi −1 + Fi −2; if i ≥ 2 Notice that starting with the third number of the sequence, a given Fibonacci number can be calculated as the sum of its two predecessors in the sequence. This means that the rst numbers can be listed as illustrated by the following table: i 0 1 2 3 4 5 6 7 8 9 10 11 12 13 : Fi 0 1 1 2 3 5 8 13 21 34 55 89 144 233 Date: January 27, 2013. 1 2 ROGÉRIO THEODORO DE BRITO This is a good moment to remark that some authors dene the Fibonacci numbers starting with index 1, instead of what we did here. As the reader will shortly see, the way we dened these numbers has advantages that will be obvious soon, while, at the same time, describing the sequence in a slightly more general way (we will also see further generalizations soon). 2.1. Some Basic Properties of the Fibonacci Numbers. A little observation of the rst few numbers listed in the table above suggests that the sequence might satisfy some relations. For instance, summing the rst four terms of the sequence gives 0 + 1 + 1 + 2 = 4 = 5 − 1; summing the rst ve terms gives us 0 + 1 + 1 + 2 + 3 = 7 = 8 − 1. This seems to suggest us that summing the rst n terms of the sequence results in the (n + 2)-th term minus one, and this is indeed the case, as we will prove in the following proposition. Proposition 1. For each natural n ≥ 1, we have (1) F0 + F1 + ··· + Fn−1 = Fn+1 − 1: Proof. Since the sequence is dened by a recurrence relation, the most straightforward way of proving it is by induction on n. If n = 1, then the left side of the equation would consist of F0 = 0 = 1 − 1 = F1 − 1, and the fact holds for n = 1. If n = 2, then we would have F0 + F1 = 0 + 1 = 1 = 2 − 1 = F3 − 1 and it is also OK for this case. The reader is encouraged to verify the assertion for other values of n, to gain a little more condence on what we are trying to prove. Let us assume then that our hypothesis is valid for all integers n such that 1 ≤ n < n0 0 0 −1 Pn 0 and let’s prove that it is also true for n . By our induction hypothesis, k=0 Fk = Fn +1 − 1. Summing Fn0 to both sides of this equation gives 0 Xn Fk = Fn0 + Fn0+1 − 1 = Fn0+2 − 1; k=0 where the last equation is simply a use of the denition of the Fibonacci numbers and this concludes our proof. Other facts can be easily inferred from the table of our rst numbers: Proposition 2. For any natural n ≥ 0, we have: (2) F1 + F3 + F5 + ··· + F2n+1 = F2n+2 (3) F0 + F2 + F4 + F6 + ··· + F2n = F2n+1 − 1 Proof. Like the previous proposition, both facts here can be proved by induction on n. We now proceed to ... In fact, “merging” the formulasDRAFT stated in Proposition 2, we can derive Proposition 1 as a corollary, obtaining, thus, a second proof for the sum of the rst Fibonacci numbers. Alternate proof of Proposition 1. Since we already know that F1+···+F2n+1 = F2n+2 and that P2n+1 F0+F2+···+F2n = F2n+1−1, summing both equation, we get k=0 Fk = (F2n+1−1)+F2n+2 = F2n+3 −1. Changing the variables conveniently, we get the result stated in the corollary. 2.2. The Golden Ratio and Some Elementary Facts. Before talking more about the Fibonacci numbers, we take some moments for a little digression regarding some other numbers that are quite important and, as we shall see, they present strong (and, sometimes, remarkably surprising) connections with the Fibonacci numbers. p 1 5 Denition 1 (The Golden Ratio and Its Conjugate). The real number ϕ = + is called p 2 ˆ 1− 5 the golden ratio and the number ϕ = 2 is called the golden ratio conjugate. ELEMENTARY RESULTS ON THE FIBONACCI NUMBERS 3 Both numbers that we have just dened occur frequently enough (including in others areas like Geometry and Architecture, for instance) that they have received attention of many authors and, as a consequence, dierent terminology is in use regarding such numbers. For instance, ϕ is also called the “divine proportion” (among other names) and the number ϕˆ is, sometimes, called the “silver ratio”. One rst elementary result involving ϕ and ϕˆ is that they are the roots of the polynomial p(x) = 1 + x − x 2 (and this can be easily seen by the quadratic formula). This means, in particular, that they satisfy x 2 = x + 1 and, in other words, we have the following fact: Proposition 3. Both the golden ratio ϕ and its conjugate ϕˆ satisfy the equality x 2 = x + 1. 2 p 2 p p Proof. We have ϕ = (1 + 5)p =4 = (1 + 2 5 p+ 5)=4 = (6 + 2 5)=4. Simplifying this latter equation, we have ϕ2 = (3 + 5)=2 = 1 + (1 + 5)=2 = 1 + ϕ2. The same can be veried for the conjugate of the golden ratio, which we leave as an exercise. In other words, if we wish to calculate ϕ squared, we can do that simply by adding 1 to ϕ. The same is true for ϕˆ. In fact, ϕ has some other interesting properties: Proposition 4. If ϕ is the golden ratio, then ϕ − 1 = −ϕˆ. p p p Proof. This is a quite direct fact, since ϕ −1 = (1+ 5)=2−1 = (−1+ 5)=2 = −(1− 5)=2 = −ϕˆ. Proposition 5. If ϕ is the golden ratio, then 1=ϕ = −ϕˆ. p 2 1 5 p p Proof. Since 1=ϕ = p · −p , we have 1=ϕ = 2(1 − 5)= − 4 = −(1 − 5)=2 = −ϕˆ. 1+ 5 1− 5 From the Proposition 5, we can formally state the same fact in equivalent, but convenient ways for our future uses: Corollary 6. For the golden ratio ϕ, we have ϕϕˆ = −1. Equivalently, we have 1=ϕˆ = −ϕ. Proof. Both facts are simply re-statements of the result of the earlier proposition. And we can, actually, say a little more about the golden ratio, as the following proposi- tion asserts: p Proposition 7. For the golden ratio ϕ and its conjugate ϕˆ, we have ϕ −ϕˆ = 5 and ϕ +ϕˆ = 1. Proof. Again, both facts admit quite straight verications. 2.3. Generating Functions and the Fibonacci Numbers. It is a fortunate case that many sequences may be “compactly” represented by a single, “simple” univariate function, whose Taylor-Maclaurin expansion (around 0)[5] has the i-th sequence number as the coecient of the i-th power of the variable x. Functions that have such properties are called generating functions [8]. It should be made explicit hereDRAFT that these functions are formal series and we are not par- ticularly concerned about convergence. If the function actually happens to be convergent for some values of x, then we can possibly use this fact for deriving other properties, but our main purpose is to just nd a function that has the chosen numbers as the coecients of the powers of the variable.1 For instance, we can easily nd a generating function the sequence h1; 1;:::; 1;:::i, that is, the sequence composed of ones only: its generating function is f (x) = 1 + x + x 2 + ··· + x i + ··· = 1=(1 − x). The reader not used to generating functions may be wondering what is, the, so special about generating functions that makes it deserve our attention. In fact a slight modication of the generating function f (x) can give us some 1In other words, we are mainly interested in the ring C[[X]] of formal series with complex numbers as the coecients. 4 ROGÉRIO THEODORO DE BRITO not so trivial results, as we shall see.