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Math Newsletter: Fibonacci Numbers

Has anyone not heard of Fibonacci numbers? They’re found in nature, literature, movies, and well, they’re famous. They’re also on the Internet, so if you really want to delve into the subject, just go online. There, I imagine, you’ll get the official version. In this article, you’ll get mine. Some resemblance should be expected and would not be coincidental – after-all, all the characters “living or dead” are all dead.  Dead since the 11th Century. So where did these numbers come from? Answer: rabbits; pairs of rabbits.

Fibonacci was tackling the problem of rabbit propagation. I can’t honestly say for sure whether he was interested in the problem ecologically or as a math-puzzle. Sources I’ve read are conflicting and Fibonacci possessed a middle-age mind – chronologically and historically. Regardless, here’s how we’ll proceed: (1) Rabbits go from baby pairs to adult pairs in one month, (2) an adult pair can conceive a baby pair after one month of adulthood (3) Rabbits live forever. All reasonable assumptions! 


Rabbit Pairs

























The pattern that emerges is that of any three consecutive numbers, the last is the sum of the prior two. 

Or, speaking mathematically: Fn+1 = Fn-1 + Fn. The first 35 Fibonacci number are listed below:

Fibonacci Numbers

F1 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12
1 1 2 3 5 8 13 21 34 55 89 144
F13 F14  F15          F16 F17 F18 F19 F20 F21 F22 F23  F24
233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368
F25 F26 F27 F28 F29 F30 F31 F32 …F35  
75025 121393 196418 317811 514229 832040 134629 2178309 9227465

What makes these numbers so intriguing to mathematicians are the amazing patterns they possess. I have my favorites. The first can be Googled, but the proof is my own, the second I discovered myself, and the third – the most magical one, is my twist on well-known results.     

The first pattern is known as Cassini’s Identity. Look at any three consecutive Fibonacci numbers, for example, 13, 21 and 34.   Square the middle one (212 = 441) then multiply the outer two by each other (13 x 34 = 442). That 442 and 441 differ by one is no chance result – it always is the case.  Try any three pair yourself.

The second pattern has no formal name but is just as interesting. Identify the position of any Fibonacci number, say F7 = 13, then look at the Fibonacci number in the doubled position – in this case F14 = 377.

Turns-out that 13 divides 377. (377 = 13 x 29). Not only is this true for any doubled-position, it is true for any position multiple. That is, F21 = 10946 = 13x 842 and F28 = 317811 = 13x24447. Put another way, if a number divides the position of a Fibonacci number then the corresponding Fibonacci numbers divide as well. Let’s look at F35  = 9227465. As 5 and 7 divide 35 then 5 (F5 = 5) and 13 should divide 9,227,465 and sure enough they do, 9,227,465 = 5 x 13 x 141,961

But we’ve saved the best for last. Recall the Golden Ratio: the value of x where x/1 = (x + 1)/x. We talked about this in a previous newsletter where we found x to be 1.61803… Now pick two consecutive Fibonacci numbers, here big is better. Say F31 and F32, 1,346,269 and 2,178,309.  Divide the former into the later: 2,178,309 ÷ 1,346,269 = 1.61803…. Now if that’s not pulling a rabbit out of a hat, what is?

We wrap things up with a proof of our result. No harm in skipping if you’ve read enough already.

Here goes:


Fn+1   =   Fn  +  Fn-1

Fn+2   =   (Fn-1  +  Fn)  +  Fn  =  Fn-1 + 2Fn

Fn+3   =   (Fn-1 +  2Fn)   +  (Fn-1  +  Fn)    =  2Fn-1  + 3Fn


More generally, Fn+K  = Fk Fn-1 + Fk+1 Fn


If k = n, we have F2n = Fn (Fn-1 + Fn+1), proving that F­n  divides  F2n

For F(a + 1)d  =  Fd + ad   =  Fad Fd-1  +  Fad + 1 Fd  -> if Fd divides Fad then it divides F(a +1)d

As Fd divides Fd , by mathematical induction we have our proof.


Going in reverse: 

Fn-2 = Fn  – Fn-1

Fn- 3 = Fn-1  – Fn-2  =  Fn-1  – (F­n  – Fn-1)  =  2Fn-1  – Fn  

Fn-4  =  Fn-2  – Fn-3   =  (Fn  – Fn-1) –  (2Fn-1  – Fn)  =  2Fn  – 3Fn-1


In the event k is even: Fn-k = Fn Fk-1  – Fk Fn-1

And in the event k is odd: Fn-k  = Fk Fn-1  – Fn Fk-1

Hence if k = n – 1 à 1 = F1 = Fn Fn-2   +/– Fn-1 Fn-1 giving us Cassini’s formula.   


Our last result follows from the observation that Fn+1/Fn = (Fn + Fn -1)/Fn  = 1 + (Fn-1/F n)  = 1 + 1/(Fn/Fn-1).

Letting n go to infinity and assuming convergence (the proof of which is out of our league), we let x represent the limiting value giving us:  x = 1 + 1/x  = (x + 1)/x, the defining equation of the Golden Ratio.