Recurrent Sequences and Difference Equations

[@blackpenredpen]

Plus, you have a great sense of humor!!!

[@H2CO3Szifon]

> "In my opinion, natural numbers start from 1" > "the first term is a_0" :D

[@sansamman4619]

this channel isn't on science and technology! it's on entertainment I feel entertained!

[@jeromesnail]

I know that it works and all, but to me it still misses at least a step (like in the bprp video about the Fibonacci sequence) to justify why we use this method...

[@angelmendez-rivera351]

jeromesnail Your feeling is unfounded. The method was justified in the video, although, to be honest, it does not require justification: if it were incorrect, then the result would simply be nonsensical and there would be contradictions. Remember that from a falsehood you can derive anything and everything, including contradictions.

[@tedhorton6315]

@jeromesnail it is based on Linear Algebra. Analogy: When a second order homogeneous differential equation with constant coefficients is solved, the method you learn in differential equations is also based on linear algebra. The equation can be rewritten as two first order equations. It is a linear system that can be represented bya 2x2 matrix. It can be solved by finding eigenvalues and eigenfunctions. The complex exponentials are the eigenfunctions. The complex coefficients of x are the eignevalues. In the case of the difference equation here, a similar thing happens (not shown in video). The lambdas are the eigenvalues for the equation, and the lambdas raised to power n are the eigenfunctions. It is important that the difference equation is written so it equals zero (homogeneous). It is second order (two eigenvalues, maybe not unique); and it has constant coefficients. Maybe you won't see this because it is very old video, so hopefully someone else can benefit.

[@anqiwang5755]

sir you just literally saved my life. tysm and i wish you the best :D

[@PapaFlammy69]

:3

[@Zzznmop]

Most lit maffs channel on the choob my boois

[@edwinlin7348]

Great video as always!! I've never actually heard of these before, or their similarities to DiffEqs, so it was very cool. Thanks a lot for sharing I just wanted to ask one question to clarify since I am a very poor excuse for an undergrad math major lol, but when you constructed the G.S. did you get that format from the way you wrote out a_n = (lambda)^n * a_0 when you first started the First order linear difference equation at the beginning? (around 4:48) I'm guessing it's same as diffEQs since the lambdas here are like the solutions to a characteristic polynomial.

[@angelmendez-rivera351]

Yes, it's rather easy to prove. The shift operator S[a(n)] = a(n + 1) is a linear operator, and so are iterations of it. Polynomials of linear operators are linear operators, a well-known fact of linear algebra, so linear combinations of found solutions to a difference equation are the general solution to said equations.

[@zubiiiiii_]

Hey I found another way to solve it. It requires thinking a bit outside of the box. Lets draw x axis, and mark 2 and 7 on it. See that (n+2)-th term is an arithmetic mean of (n+1)-th and n-th. Therefore we will have to kinda move in the middle of those two points and so on, so on. We started at 2, then we move 5 units forward, 2.5 units back, 1.25 units forward..... so our final distance from origin is: 2 + Sum from 0 to infinity of (5 × (-1)^n / 2^n). We can evaluate this sum by using geometric progression, with 1st term of 5, and ratio of -1/2. By using a/1-q we get our infinite sum equal to 10/3, and by adding 2 from the beggining, we get final answer of 16/3

[@muhammadsyifaurrohman2314]

wow, good job! :0

[@deceo2119]

Thank you past Jens, you just helped me for an upcoming Linear Algebra exam!

[@philippnirnberger4708]

Thank you for this video. You really helped me understand recursive difference equations. Is there also content about recursive functions with a disturbance like: 4x(n+2)-8x(n+1)-5x(n)=1-27n. Thank you for the great content!

[@holyshit922]

First way Generating functions (ordinary and exponential) Second way (for linear constant coeff) Rewirite homogeneous part of equation with system of equation and solve it with algebra (calculating matrix power with some matrix decomposition , eigenvalues and eigenvectors can be useful) Partial solution for inhomogeneous equation we can find using variation of parameter It is similar to differential equations but we have Casoratian instead of Wronskian and summation insead integration

[@cherylann1104]

Very nice video and 2nd order linear difference equations very well explained - you simplified it soooo well ! Thank you

[@jihanhamdan5465]

Yeahhhhh pleaseeee dooo moreee on difference equations... Please explain starting from 0, different cases and stuff like that. U said they're like differential equations. There we have homogeneous and non homogeneous and these stuff. So if u please introduce us to the types of difference equations also... And thanks to blackpenredpen for the video he did with u so i was introduced to ur channel too. A big horaaaaay for him... Oreoooo.

[@dinoanastasopoulos8511]

Referring to 3:22, is this equation not 2nd order? Rather than 1st order as you said

[@yethusithole4695]

C_{1} =16/3 and C_{2}=20/3, there's a mistake you took 5X(4/3) and said it's a mixed fraction on your solution for C_{2}.

[@SugarBeetMC]

0:20 You mean there's nonzero interest in this topic.

[@DarkRedZane]

For all epsilon > 0

[@Nicholas-gr5pb]

This is a question from last year’s NCEA Scholarship Calculus Paper!!!

[@katlehomolete9961]

Thank you soo much☺

[@FruitloopLeviathan]

5:52 why do you have the right to cancel the a0?

[@shandyverdyo7688]

Instantly I realized that the past flamm was softer than now. Kidd jens. ✌ :'v

[@legasa6343]

superb, I like this

[@PapaFlammy69]

:)

[@caio868]

Doing econometrics of 1 data point haha! Thank you, really good.

[@ethanjensen661]

This is amazing!

[@thobi6639]

Thank you

[@jimboxb]

Why would you not include zero in the natural numbers? We already have a name for the set {1, 2, 3, ...}. The positive integers.

[@ItzGrundy]

James Barnes It's often not considered because you can't have zero of something. (It's not natural lol)

[@angelmendez-rivera351]

ItzGrundy Except you CAN have zero of something. It is called the "trivial" case, because you always have at least zero of something in magnitude. If a trivial case can exist, then it must exist.

[@soumyadipsarkar7242]

😂😂😂😂😂the dog is really nice..

[@katlehomolete9961]

Would I still be correct if I said lambda 1 is equal to negative half instead of 1?

[@fidgetspinner1050]

more sequences!!!

[@Witiok1992]

I would like to have a tutor like you at my university.

[@davidmendizabal9892]

i think natural numbers strat with one (because we need some reason to start counting)... but then inmediately follow the realization that not counting is in itself a form of count (because you now are aware that before you wasn't able to count, as there was nothing to count) so the distinction between now that you have counted 1 and before that nothing was being counted deserve its own symbol. so then the natual numbers are as follows: 1,2,3... oh shit! no, no, no! 0,1,2,3,4,5,6,7...

[@cedricp.4941]

your dog is so cute 😍

[@cedricp.4941]

both 😊

[@Witiok1992]

What do you think about math education in Germany. I mean schools.

[@soumyadipsarkar7242]

Great video,,man!!!!!!,but u did not clarify how to recognize a homogenous equation..

[@Witiok1992]

In which field will you work after university?

[@rayeesashraf5513]

Want more such videos

[@nisaragbhatt1098]

Thank you so much!

[@milanmilacic9311]

I hated this topic because at my uni we don't really have proper maths literature we use notes of previous students and particularly for this topic we were just given the algorithm without the explanation as to why that's the case, so thanks

[@waynelast1685]

whats the goal

[@cycklist]

Tiny indices in low-res and bad lighting are tricky to read :-(

[@edwinlin7348]

I think it was fine lol it was clear the indices were from hearing you say it. And they were self-explanatory for the most part. Don't sweat it too much :D

[@doggo4440]

He’s so handsome 😍

[@aneeshsrinivas9088]

couldn't you just do a discrete version of variation of paramaters.

[@moslemasultana9388]

how a(n)=c1lemda1^n+c2lemda2^n?

[@tsunningwah3471]

hi flammy, just wonder why you knew An+1=入An in first place

[@brydust]

Math hack: Be lazy, express c_2 in terms of c_1, solve for c_1 and forget about c_2 since that terms vanishes. [at least helpful on an exam if you only care about the limit]

[@Djake3tooth]

1:15 the fact you use 0 as first index but not as a natural number confuses me

[@soumyadipsarkar7242]

U simply assumed the sequence to be homogeneous,,but how can u do that??

[@angelmendez-rivera351]

Soumyadip Sarkar Look at the damn equation he wrote and tell it is not homogeneous. Seriously. Use your brain cells.

[@newkid9807]

Angel Mendez-Rivera Mendez stfu.

[@davidseed2939]

You didn’t make the important point that the value in the limit is (a1+2a2)/3

[@HoaNguyen-no5ob]

a video is dark, u should make it brighter

[@oscardavidalarcon2673]