I didn't know this method it is really smart. You can also using the fact that a = (1+√5)/2 and b = (1-√5)/2 are both solution of the equation x² = x + 1, hence a² = a + 1 and b² = b + 1 You can deduce the following: a³ = aa² = a(a +1) = a² + a = a + 1 + a = 2a + 1 a⁴ = aa³ = a(2a + 1) = 2a² + a = 2a + 2 + a = 3a + 1 And so on, hence a^n = U[n-1]a + U[n-2] and same thing for b, b^n = U[n-1]b + U[n-2] By substracting the two expressions: a^n - b^n = U[n-1](a - b) (the U[n-2] cancel each other) Hence U[n - 1] = (a^n - b^n)/(a - b) where a - b = √5.
You should mention this in the net part, because this video feels incomplete without it. You, as a teacher shouldn't just pull the variables a and b out of thin air, instead of giving some insight on why you chose them.
Sure there are many ways to deduce it, the very fun is to go through every one and to compare them, and to watch how every branch of math deals with the same thing, through its own mechanism and interpretation...
Although the numbering of the terms in the Fibonacci sequence is essentially arbitrary, there are various reasons why it is preferable to set F[0] = 0 and F[1] = 1. First, it simplifies the formula for the nth term, resulting in Binet's famous formula, which is expressed with powers of n rather than n-1. Secondly, it keeps it in line with the Lucas numbers, in which L[0] = 2, L[1] = 1, L[2] = 3, etc., for which there is a similar (slightly simpler) formula, also with powers of n and not n-1. Thirdly, it allows the observation that if F[n] is a prime greater than 3, then n is prime. Fourthly, it allows simple formulae for F[2n], F[3n], etc. E.g., F[2n] = F[n]·L[n], or F[2n] = F[n]·(F[n-1] +[F[n+1]). Fifthly, both the Fibonacci sequence and the Lucas numbers can be extended backwards, resulting in a symmetry (ignoring signs) centred on n = 0. I'm sure there are many other reasons why the two sequences have the particular numbering that they do.
The cool thing about this is that you can easily generalize the idea of fibonacci numbers and instantly get the result for more insane recursions (as long as they are linear) like an+1 = an + an-1 +an-2 +...+a0 which is basically the same Problem (Just calculate the Eigenvectors and Eigenvalues)
this is very neat ! still waiting for the proof that pi and e are Transcendental numbers also, it would be expected but because this is material that i already study this video wasn't the most entertaining for me, i like the "heavy" stuff you do more
It’s definitely on my to-do list, but don’t expect a video anytime soon, there’s a bunch of other stuff coming ahead. And I understand, but since there’s a mixed audience, I like to alternate between elementary stuff and advanced stuff; otherwise only a handful of people would be able to follow my presentation :)
Dr. Peyam's Show so I'll just send a reminder every video till then XD I understand the last part, I am not complaining, just saying what I like more. Keep it going
unfortunately the generalised n-fibbonnaci sets all values from f0 to f(n-1) to 1 instead of having zeroes occupy everything prior to fn as a programmer would generalise it
Created for counting bunnies Me:why am I not surprised?oh yes because most theories and discoveries were created in simple ways gravity by an apple falling as an example it's incredible and surprising
This way would be nice if he showed some matrix decopositions which help to raise matrix to the power but still generating function looks better for me 8:10 This matrix is diagonalizable but generally diagonalization is not enough
Quality work! I think I probably saw this when I first was learning linear algebra, but I certainly don't remember the initial set up to solving it as a LA problem. How would you do a series like X(n+1)=X(n)^2+1 using linear algebra? I think I've seen Fibonacci formulated using generating functions/kind of like power series expansion and analysis of the resulting functions terms and maybe one other way. Obviously you can do a proof by induction once you have the seed of an idea, but the methods you'll probably cover will be generic to some extent.
Sadly linear algebra only works using linear recurrence relations, so your example wouldn’t work! Also the next part will be like the way you’re mentioning :)
Is there a video of someone using the binomial theorem to show that this is always a whole number and getting it in sigma notation instead if in terms of square roots of 5?
In which step of this calculation does the formula narrow down to the fibonacci sequence? Because the beginning would seem to be a general foundation for all sequences of the form Un+1 =Un + Un-1 that don't only start with 1,1
Hey! I've recently discovered your channel and I find these discussions so eye opening. I'm currently taking AP calculus as a senior and we just started getting into definite integrals they are very confusing ,and I would like to know where I should start learning on the subject? Thanks!
