The Fibonacci Sequence and Experies with Learning Nate Young TEDxYouthConejo

before I start I just want to say this is going to be a lot of math and if math tends to put you to sleep I'm sorry and sweet dreams but what I want to talk about today is as you can see the Fibonacci sequence and I want to talk about a few of my experiences with it and what those experiences have led me to believe what they have let me learn about learning itself and about the learning process now many of us here probably know what the Fibonacci sequence is if you don't here it is now you do congratulations a lot of us know how to get the Fibonacci sequence how its defined if you don't here it is now you do congratulations it's the sequence that begins 1 1 or 0 1 depending on who you ask and continues where every number is the sum of the two before it so 0 plus 1 is 1 1 plus 1 is 2 2 plus 3 is 5 etc and what I what I think about learning this is the first this is the first learning point I want to touch on is if you want to learn something you have to be interested yourself you have to be motivated to learn you have to be intrinsically motivated to learn and there's no end of reasons why the Fibonacci sequence is interesting and why people would want to learn about it there are a lot of mathematical properties for the Fibonacci sequence here's the first one I want to talk about Pascal's triangle is used to to expand binomial so if you have X plus y cubed you can see that the the four numbers on the bottom there 1 3 3 1 those correspond to a certain row of Pascal's triangle and that's what it's used for but if you take the shallow diagonals of it as you can see in the picture those those shallow diagonals if you add up all the numbers they will give you the numbers at the Fibonacci sequence another mathematical property the cassini and Catalan identities the Ksenia he says that for any three consecutive terms in the Fibonacci sequence say three five eight if you multiply the outer two and then you square the beginner one they will differ by exactly one so 3 times 8 is 24 5 squared is 25 they will differ by exactly one no matter what three consecutive terms you pick Zeckendorf theorem says that any natural number you can think of any natural number can be arrived at by adding up different Fibonacci numbers without repeating any so say you want to make a hundred eighty nine plus eight plus three 102 89 plus eight plus three plus two etc any any number can be arrived at in this way and in addition to these mathematical properties which certainly interest me a lot of people are interested in the Fibonacci sequence for their natural properties the seeds in the petals of flowers here you can see that they're arranged in a way that can be modeled by the Fibonacci sequence and using a few of its properties that's the model right there in the bottom left the branching of trees and leaves if you have a tree oftentimes the number of branches at a certain level will correspond to the Fibonacci sequence and if you have say a certain type of plant that has has eight leaves on it to go around in a spiral oftentimes that spiral will wrap around the plant itself five times and it's common for for these to both be Fibonacci numbers I skipped one but there it is in nature this is actually this is actually how Fibonacci originally described the sequence with the breeding of these idealized rabbits there's one pair of rabbits and then they mature and they produce another pair who in turn mature and produce another pair and these are of course very idealized none of them can ever die all of them produce exactly one every every one pair every turn they all produce the they all produce more rabbits at exactly the same time and you run into all kinds of entertaining genetic diversity problems but it's interesting that this is the way it was originally described and then it works like this for a number of different animals and finally pineapples and pinecones the spirals of them you can see Fibonacci numbers and those for example in the you know in the number of different ways that you can see a spiral and a pineapple or the number of spirals that you can see on a pinecone but the main the main fibonacci property that my talk is focusing on is this one Phi Phi is a number it's one point six one eight or there abouts it's an irrational number and it's one plus the square root of five over two and the reason it's important is because it's the solution to both these equations x squared equals x plus one and one over x equals X minus one and it is also the converging ratio of the Fibonacci sequence so if you go out to the infinity of number of the Fibonacci sequence it's a never-ending sequence but if you take an infinity at number and you divide it by the one before it you will be left with fine and that's what really interested me is that this number which is interesting in its own right which is important in many branches of mathematics and which seems to be the human standard for aesthetics is such an important number to such an important sequence and here's where I want to talk about my experience because what I did once was I wrote a program on my calculator to get Phi given the Fibonacci sequence and there's this very interesting graph that it traces about where where the ratio gradually approaches fine but then I turned it around and instead of multiplying instead of dividing the numbers I multiplied them and in that case you are left with a completely different sequence since you're multiplying the consecutive terms and it's 1 times 1 is 1 1 times 2 is 2 2 times 3 is 6 etc and what really struck me about this was that the converging ratio of that sequence is Phi squared and if you do it again it's Phi to the 4th and then again its Phi to the 8th and if instead of multiplying two terms together you multiply three it's fight of 1/3 and then fight ix and that's what i thought was interesting because in effect this is like squaring the sequence you're not actually squaring the sequence but it's similar and the converging ratio is squared now it turns out that this in itself is not particularly profound the fact that this is true for any sequence with the conversion ratio is short enough that you can fit it on a PowerPoint slide and I did but this is the first of three major learnings through major discoveries that I that I had visited just the first of them and what I want to say is what if what if I haven't come up with this on my own what if we had been taught this in school what if I had taken some class and they had either shown us this proof or you know even just given it to us which is common or even if we had gone through it step-by-step in class where I've continued what I have learned as much as I did what I have been interested in it I don't think so I think that in if you want to really learn something and if you want to learn a lot you have to be interested you have to be interested in it of your own accord the second thing is that the sum of the squares so let's say you square the Fibonacci sequence 1 1 4 9 25 64 and you take the sum of all of the numbers in that sequence up to a given point you'll be left with 1 2 6 15 40 104 that's the same sequence as if you multiply consecutive terms together and that's something that only happens with a Fibonacci sequence and this is actually well known this I could have been taught in the classroom if I had taken a class on mathematical sequences are on the Fibonacci sequence itself I probably would have been taught this but it's because I wasn't that I was able to really understand the thinking behind it and the way it works is because I was interested in to myself and I was intrinsically motivated but the last one and this is where where I really found something interesting is an explicit formula explicit formula is a something that use the number of the term so for the Fibonacci sequence it's based on the two before it if you need to find the hundredth term you need to have the 99th and the 98th if you need to find the 99th you have to have to have the 98th and the 97th and you can see where this is going an explicit formula all you need to know it is that it is the term 100 and there is an explicit formula for the Fibonacci sequence it's called Benes formula and what I did is I said about using Benes formula to have an explicit formula for any of these sequences any sequence where it's where it's been multiplied together and there it is it's it's rather long but it's it works and it's for these for these sequences a which you see up at the top there is the number of times that you've multiplied these terms together so if it's 0 is just the Fibonacci sequence if it's 1 you multiplied it once and it's 2 twice and you can see that the the converging ratio is followed that pattern but the real real interesting thing the really interesting thing that I want to touch on is if you look at the the exponents of the numerator there they follow this pattern it it's lowercase a or whether it's a in a subscript capital C 1 and a subscript capital C 2 etc and the reason that that is entertaining is that as the eighth row or row a of Pascal's triangle and in a way that's interesting because in a way it makes sense usually for Pascal's triangle you're adding it together a bunch of terms and each term is multiplied by a coefficient that you're given by Pascal's triangle but here you're multiplying together a bunch of terms and each of them is raised to a power and you're given that power by Pascal's triangle and this is not something I would have expected to see here even though it makes sense I never expected to find it here and that's why I say that if you learn enough and you're intrinsically motivated to learn and you're interested and you learn you learn enough and you keep learning you might just learn something that you already knew thank you

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