What interesting properties of the Fibonacci sequence can I share when introducing sequences?

Question

The Fibonacci numbers are one of the first sequences given as examples of sequences in many calculus textbooks as they have a definition that does not obviously have a closed form and they have many real-world applications.

But that last part has been very controversial in recent years, with people maki g extravagant claims for and against its appearance in nature.

What claimed properties of the Fibonacci sequence (mathematical or in real life) are both factual and interesting to first year calculus students?

Answer 1

• Since $\varphi$ is rather close to the conversion rate between miles and kilometers, one can use the Fibonacci numbers to convert: if $f_n$ is the distance in miles, then $f_{n+1}$ is (roughly) the distance in kilometers. You can use this to facilitate a discussion about, first of all, the convergence of these ratios $\frac{f_{n+1}}{f_n}\to\varphi$, and also about how the accuracy of the conversion gets better with larger $n$. What is the rate of convergence of the ratios? I think students will find these discussions interesting.
• Summation properties of the Fibonacci sequence can serve as quasi-interesting facts for the students, but they can also serve you as a way to gently introduce them to induction without really spending an entire unit on it. For instance, work with the students to first observe and then prove that $\sum_{k=0}^n f_k = f_{n+2}-1$. Discuss your proof and why it's necessary, why we can't just observe the pattern and say, "Oh yeah, it keeps going, trust me ..."
• If you have some students with computer science interests, you can talk about Zeckendorf normal form of integers, wherein you encode an integer by expressing it as a sum of Fibonacci numbers. Amazingly, every integer has a unique such representation where no two consecutive Fibonacci numbers are used! Without the consecutivity, existence is trivial to prove but uniqueness is not guaranteed. This topic can facilitate a discussion of existence and uniqueness in mathematics, in addition to considerations of computational efficiency.
• This combinatorial idea is also mentioned by Noah's answer, but I'll modify it slightly and expand: $f_n$ is the number of ways to tile an $n\times 2$ chessboard with dominoes. You either take a tiling of an $(n-1)\times 2$ board and append a vertical domino, or take a tiling of an $(n-2)\times 2$ board and append two horizontal dominoes. This kind of simple counting argument can be followed by calculus students, and they might see how these kinds of sequences come from natural questions about "How many ways are there to do $X$ with property $Y$?" You could even expand on this and point out the appearance of Fibonacci numbers along the diagonals of Pascal's Triangle:
• Without getting too deeply into the general ideas, you can present the generating function for the Fibonacci numbers. You could hold off on this until you talk about power series later on in the course, or later in the calculus sequence.

Answer 2

I think the Fibonacci numbers should be mentioned in a first-year calculus class, though they should not be discussed at length. I wouldn't try to stress the real-world applications—these range from convoluted to dubious, and aren't really the reason that Fibonacci numbers are famous.

Here are some good reasons to mention them:

• They are a famous sequence, and being aware of them is part of general mathematical literacy.

• They provide an excellent example of a recursively defined sequence, another topic that deserves to be mentioned. (My other favorite recursively defined sequence is the seqeunce defined by $a_1 = 1$ and $a_n = \cos(a_{n-1})$, which converges.)

• They are related to many of the ideas that come up during the sequences & series unit. Indeed, there are several places where Fibonacci numbers provide an interesting example of the material under discussion.

Here are a few examples supporting the last point.

Question. Does the series $\displaystyle\sum_{n=1}^\infty \frac{1}{f_n}$ converge?

Answer: It turns out that $f_n$ goes to infinity geometrically as $n\to\infty$. In particular, $$ \lim_{n\to\infty} \frac{\varphi^n}{f_n} = \sqrt{5} $$ where $\varphi$ is the golden ratio. Thus the series converges by the limit comparison test.

Alternatively, you can use the ratio test since $$ \lim_{n\to\infty} \frac{f_{n+1}}{f_n} = \varphi $$ or the root test, since $$ \lim_{n\to\infty} \sqrt[n]{f_n} = \varphi. $$ The advantage of bringing this up is to get the students to think about geometric growth a little abstractly. That is, the Fibonacci numbers provide a nice example of a sequence that grows geometrically, but for which this isn't obvious. I wouldn't attempt to prove any of these statements, I would just state them as empirical facts, and then make the connection to convergence of series.

Question. What's the radius of convergence of the series $\displaystyle\sum_{n=1}^\infty f_n x^n$?

Answer. It's $1/\varphi$. There are plenty of other interesting radius of convergence problems that involve an $f_n$ somewhere in the formula for the coefficient. Again, the purpose of mentioning this would be to get the students to think about the relationship between the coefficient of $x^n$ and the radius of convergence.

Question. What's the sum of the series $\displaystyle\sum_{n=1}^\infty f_nx^n$?

Answer. There's a neat trick for this. Let $$ f(x) = x + x^2 + 2x^3 + 3x^4 + 5x^5 + 8x^6 + \cdots $$ Then $$ x f(x) = x^2 + x^3 + 2x^4 + 3x^5 + 5x^6 + 8x^7 + \cdots $$ and $$ x^2 f(x) = x^3 + x^4 + 2x^5 + 3x^6 + 5x^7 + 8x^8 + \cdots $$ By inspection $$ f(x) \;=\; x + x f(x) + x^2 f(x) $$ and solving for $f(x)$ gives $$ f(x) \;=\; \frac{x}{1-x-x^2}. $$ I like this trick, because it's a nice variation on the usual trick for finding the sum of a geometric series.

Answer 3

Mathematics

(People have already mentioned many of the great properties of this sequence at the time of answering. These are a couple of good ones that were missed.)

Combinatorics

The Fibonacci sequence is the number of strings of characters of length $n$, containing only a and b, where two a's cannot be consecutive. For $n=4$ there are $8$ strings for example:

abab abba abbb baba babb bbab bbba bbbb

Relation

Another nice factoid is that $F_{n-1} F_{n+1} = F_n^2 + (-1)^n$.

This relation, and lots of other nice ones, including the limiting behavior, follow from the closed form $$ F_n = \phi_1^n + \phi_2^n $$ where $\phi_1$ and $\phi_2$ are the two solutions to $x^2 - x - 1 = 0$ (and the positive one of these is the usual 'golden ratio' $\phi$).

Solving the recursion to find this closed form is neat and not too complicated, although it does require fluency with algebraic manipulation, and some tricky concepts. If you can do this, you can see why all Fibonacci sequences, regardless of the first two numbers, have the same limiting ratio between two adjacent terms, and how all the closed forms are related.

If you don't want to find the closed form, you can probably still obtain the relation, just using the recursive definition.

Phi

The irrational number which can be least well approximated by fractions of a given size of denominator, is $\phi$. The best approximations at any size are given by $\frac{F_n}{F_{n-1}}$.

For most irrational numbers, if you try to find fractions which are close to that number, but whose denominator is not too big, you occasionally stumble across fractions which are much close than the 'average'. Eg $\pi$ can be written approximately as $\frac{3}{1}$, $\frac{22}{7}$, and then $\frac{355}{113}$. The last one is much, much better than you would hope for - the error is one over 5 million whereas you should statistically only expect an error of one in a few tens of thousands, for a denominator of that size.

With $\phi$ you never 'get lucky' - fractional approximations, even the best possible fractional approximations, always fall roughly the expected distance away from $\phi$. They also alternate between too big and too small.

To fully understand the math behind this, you have to learn about continued fractions, which are extremely interesting, but probably off-topic for your class. The math is not very difficult, but it's not the kind of thing you can explain in a 10 minute parenthesis.

Pseudo-mathematics

A lot (perhaps most) of what is claimed about the Fibonacci sequence and the (very closely related) Golden Ratio is bogus. A nice paper is here: http://community.dur.ac.uk/bob.johnson/fibonacci/miscons.pdf

Some pointers:

Natural world

• The Fibonacci sequence has a pretend real-world justification, in terms of rabbits reproducing. No-one sensible has ever claimed that these number were observed in real rabbit populations.
• The Fibonacci sequence does appear in some plant physiology, notably numbers of branches, petals and seeds for certain plants. There are some good reasons for this: some plants branch in simple ways which are analogous to the 'rabbit family trees'.
• Some of these patterns also involve the 'Fibonacci spiral', eg sunflower heads. This has a slightly more complex mathematical justification, which is not at all controversial.
• Lots of people have found Fibonacci numbers, the golden ratio and Fibonacci spirals in many, many natural objects. Most of these claims are either coincidence or deliberate (perhaps unconscious) mismeasurement (unless the universe works very, very differently to the way most scientists think it does).

Examining 'fringe' claims

• Some of these people are just on the wrong track, or doing something of interest to them which isn't quite mathematics or science, some of them are probably insane. It's often hard to tell papers written by these people apart from serious writing about the links between mathematics and say, zoology.
• Some of them might be right, and have spotted observed something about the mathematics of say, a zebra's stripes, that wasn't previously known to professionals. What is very unlikely is that all of these claims are true.
• The Golden ratio is probably the worst offender. If you make enough different measurements of parts of an object, you are pretty certain to find close approximations to any ratio you care to choose.
• Finding small values from the Fibonacci sequence (such as 5 or 8) is another common trick.
• Some writing which purports to find several occurrences of $\phi$ or the Fibonacci sequence in a given natural object, actually finds one such occurrence, and then transforms it in various simple mathematical ways and claims that these are independent observations.
• The ratio of the side to the diagonal of a pentagon is $\phi$. Natural objects with fivefold symmetry (some fruit, starfish) will probably contain the golden ratio without this meaning much, except that they have fivefold symmetry.

Art and culture

• Some historical and contemporary architects, artists and designers have intentionally used $\phi$ or the Fibonacci sequence. The most famous example is the Parthenon in Athens.
• Some of these people believed that these ratios and numbers were specially beautiful, some that they had some kind of mystic power, and some were playing around with interesting math.
• Some of the intentional use by artists might have degenerated into the belief that the numbers are present in physical objects. For example, artists might have chosen to use $\phi$ for the ratio of navel height to height when drawing figures, and this might have led people later to believe that this exact ratio actually occurs in all humans.
• As with natural objects, it has also been claimed that $\phi$ and Fibonacci appear in almost all buildings, paintings, sculptures and music in various disguised or non-disguised forms. Much of the 'evidence' of this works the same way as with natural objects - measure many different parts of a painting, and work out all the ratios until you find the ones you want.
• There is no evidence that our brain is wired to select these patterns unconsciously or to find them especially beautiful.
• Nor is there a scientific consensus that $\phi$ and Fibonacci are ubiquitous in all human endeavour at all times in history because of divine inspiration, secret knowledge handed down by aliens, or an extremely wide-reaching conspiracy.

If you discuss this with your students you will probably hear a lot of the bogus claims above which have passed in the popular consciousness, partly as a result of 'The Da Vinci Code'. I think examining these is great math. Depending on how focussed on your syllabus you have to be, you might be able to discuss some or all of the following with your class:

How accurate does the ratio between a person's height and the height of their belly button have to be, before we can be sure that it is $1.618$ and not $1.638$ or $1.5922$? Is the related claim that the height of the belly button and the vertical distance between the top of the head and the belly button are in the same ratio further evidence, or is it redundant? How many people should we measure? How many daisies should we count petals of to decide if only Fibonacci numbers appear, and how close to the exact numbers do we need to be? What signs within a math paper and from its context should we examine to decide if it can be trusted? Is the distinction between mainstream math and pseudo-mathematics valid or is pseudo-mathematics just mathematics by people outside the mainstream?

You might or might not be interested in talking to your class about this. I had some success when students in a Calc I class asked me about appearances of $\pi$ in Egyptian pyramids. However even if you don't want to go into it, you should say just enough when talking about this topic that your students understand that not every claim they might read is reliable.

Answer 4

If I were to cover the Fibonacci Sequence in introducing sequences to Calculus students, I would probably avoid some of the more obscure properties. I would focus on what early Calculus students should know: for example, how to prove that a sequence converges, and if it does, then how to find what it converges to.

(Note also the places, not necessarily specified below, in which one needs to remind students about the basic properties of sequences and limits, and when things can be "split up" only when convergence has already been demonstrated.)

In this case, one can show the ratio $\frac{f_{n+1}}{f_n}$ is monotonically increasing and bounded above by $2$.

This is enough to conclude that the ratio converges to some limit as $n$ tends to infinity.

Call this limit $\phi$. Then consider two ways of writing $\frac{f_{n+2}}{f_n}$ as $n \rightarrow \infty$. The first way:

$$\lim_{n \rightarrow \infty} \frac{f_{n+2}}{f_n} = \lim_{n \rightarrow \infty} \frac{f_{n+1} + f_{n}}{f_n} = \lim_{n \rightarrow \infty} \frac{f_{n+1}}{f_n} + \frac{f_{n}}{f_n} = \lim_{n \rightarrow \infty} \frac{f_{n+1}}{f_n} + 1 = \phi + 1.$$

The second way:

$$\lim_{n \rightarrow \infty} \frac{f_{n+2}}{f_n} = \lim_{n \rightarrow \infty} \frac{f_{n+2}}{f_{n+1}} \cdot \frac{f_{n+1}}{f_n} = \lim_{n \rightarrow \infty} \frac{f_{n+2}}{f_{n+1}} \cdot \lim_{n \rightarrow \infty} \frac{f_{n+1}}{f_n} = \phi \cdot \phi = \phi^2.$$

Since these expressions are equal, we find that $\phi^2 = \phi + 1$, whence we can solve for $\phi$ using the quadratic equation. This will give two possibilities: one positive, and one negative. Noting that $\phi > 0$, we find the limit to which our ratio converges.

---

On the other hand, if you are looking for a somewhat nonstandard problem:

Call a binary string "tripletless" if it never contains three consecutive $0$s or $1$s.

How many tripletless binary strings are there of length $n$?

I worked this problem out some time ago, and found the answer is $2f_{n+1}$.

For example,

Length $1$: $\{0, 1\}$. Total: $2$, i.e., $2f_{2} = 2\cdot1$.

Length $2$: $\{00, 01, 10, 11\}.$ Total: $4$, i.e., $2f_{3} = 2\cdot2$.

Length $3$: $\{001, 010, 011, 100, 101, 110\}.$ Total: $6$, i.e., $2f_{4} = 2\cdot3$.

Length $4$: $\{0010, 0011, 0100, 0101, 0110, 1001, 1010, 1011, 1100, 1101\}.$ Total: $10$, i.e., $2f_{5} = 2\cdot5$.

(If more details on the proof would be helpful then I would be happy to provide them, but I think it is a nice problem to work out. For pedagogically appropriate materials, though, I think the example above is a much better choice.)

Edit (4 April 2016): An irresistible mathematical side-note. The "somewhat nonstandard problem" here essentially asks for a count of

base $2$ strings of length $n$ without $3$ consecutive occurrences of the same digit.

A natural generalization would be to ask for a count of

base $a$ strings of length $b$ without $c$ consecutive occurrences of the same digit.

I asked this very question way back in Apr 2014 on MSE (775863) and it now has a nice answer from Markus Scheuer that uses generating functions and the "Goulden-Jackson Cluster Method" (arXiv).

Edit (27 March 2017): I recently explored with a class of mine the following:

$$1+\frac{1}{1}, 1 + \frac{1}{1 + \frac{1}{1}}, 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1}}}, \ldots $$

where each subsequent term is $1$ plus $1$ over the previous term. These yield

$$\frac{2}{1}, \frac{3}{2}, \frac{5}{3}, \ldots$$

and, more generally, one can show that you get ratios of consecutive Fibonacci numbers. So the next one would be $8/5$, then $13/8$, etc. When you really do this out by hand, e.g., observing that $1 + 1/(8/5)$ is $1 + 5/8 = (8+5)/8 = 13/8$, you not only get to grasp why the Fibonacci numbers are emerging, but by taking the "limit" of the sequence, you have that $x = 1 + 1/x$, which allows you to solve for $x$ and pick the positive root of this disguised quadratic to figure out precisely what the ratio of consecutive Fibonacci terms converges to.

Answer 5

Our math teacher came with the fun fact that the sum of the first 10 terms equal to 11 times the 7th term no matter what starting numbers you choose.

Example: 1 + 1 + 2 + 3 + 5 + 8 + 13 + 21 + 34 + 55 = 143 = 11 * 13.

The reason it works with any starting numbers is because you can rewrite it as: (a) + (b) + (a + b) + (a + 2b) + (2a + 3b) + (3a + 5b) + (5a + 8b) + (8a + 13b) + (13a + 21b) + (21a + 34b) = 55a + 88b = 11 * (5a + 8b)

Answer 6

The relation with squares is pretty nice:

$\sum_{i=1}^n {F_i}^2 = F_{n} F_{n+1}$

The video link shows how suitable this concept might be for consumption by a general audience.

Answer 7

Some number theoretic properties:

For any number $N$ there exists a Fibonacci number divisible by $N$, which also means that for any $N$ there exists a Fibonacci number ending in $N$ zeroes.

$F_n$ divides $F_m$ iff $n$ divides $m$ except for $F_2 =1$ which divides more stuff, this requires that you start with $F_0 =0$ and $F_1 = 1$.

Following a similar vein we get that $\gcd(F_n,F_m)= F_{\gcd(m,n)}$.

As a corollary $F_n$ can be a prime number only when $n$ is prime. One might ask: Are there exist infinitely many prime Fibonacci numbers? Heuristics suggest that the answer should be yes but it is an open question.

A slight variation of that with a different outcome is that $34$ is the only Fibonacci number that is twice a prime number.

Answer 8

Proofs That Count by Benjamin and Quinn has some interesting statements about Fibonacci numbers. They establish a correspondence between Fibonacci numbers and the number of ways to tile an $ n \times 1$ strip with tiles that are either $ 1 \times 1$ or $ 2 \times 1$, so every claim they present has a visual/combinatorial interpretation. A lot of the statements relate Fibonacci numbers to binomial coefficients. Here is one statement from the book: four consecutive Fibonacci numbers generate a Pythagorean triple:

$ (f_{\text{n-1}}f_{\text{n+2}})^{2} + (2f_{\text{n}}f_{\text{n+1}})^{2} = (f_{\text{2n+2}})^{2} $

The real appeal of the book for me isn't any particular statement but the correspondence the authors establish. I personally find it interesting, but I think PVAL's comment has a lot of merit.

Answer 9

Unless it shows up naturally (or it's generating function makes a good example, ...) better leave it out. There are plenty of other topics, more easily related to the main subject.

You see from the other answers that there are lots of cool combinatoric interpretations, but the properly "analytical" are too deep for a first course. Sadly. They even have their own journal, The Fibonacci Quarterly.

Answer 10

Possibly Lamé's theorem?

Here are some extra characters because the system won't let me input only the prior paragraph.

Answer 11

How about the fact that the n+1th number in the sequence divided by the nth number approaches phi? I always thought that was cool.