Fibonacci Sequence with Differential Equations

Today I thought I’d talk about something I recently did in one of my lectures.

As most of you probably already know, the Fibonacci sequence is defined in the following way:

{\displaystyle F_{n}=F_{n-1}+F_{n-2},}

with initial values  {\displaystyle F_{1}=1,\;F_{2}=1}.

This sequence comes up everywhere, for example in biological systems describing the number of petals and the shape of broccoli.

Now let us solve the equation


As this this is a difference equation, we can solve it using techniques for differential equations. First, we make the ansatz (this is just fancy German word for ‘educated guess’) that


This will give us the simple equation: Screen Shot 2016-11-10 at 8.28.26 AM.png

This, as you can probably recognise, means that k is the golden ratio and its inverse!

Screen Shot 2016-11-10 at 8.31.06 AM.png

“For reasons that have still not been thoroughly resolved by neuroscientists, we are conditioned to find aesthetic appeal in structures with aspect ratios close to the golden ratio, a fact known empirically to artists and engineers from ancient times. One everyday example is the proportion of pieces of paper.” – Dr. C.P Caulfield 

Therefore, we know that the solution to the difference equation is

Screen Shot 2016-11-10 at 8.32.25 AM.png

Using the initial conditions, we can find A and B:

Screen Shot 2016-11-10 at 8.33.03 AM.png

Note that this is an expression for a sequence of integers in terms of differences in the powers of the golden mean, which is certainly an irrational number (in fact it can be argued to be the most irrational number of all as it can be expressed in terms of the convergence properties of its continued fraction representation)!

Also, as φ1 > 1 and rearranging the above expression, we can see that:

Screen Shot 2016-11-10 at 8.35.56 AM.png

So the ratio of consecutive numbers in the Fibonacci Sequence tends to the Golden Mean as n gets very large. Isn’t that amazing!

M x

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