@ Lud R - That is the general form of a cubic equation (An^3 + Bn^2 + Cn + D). Any cubic equation can be expressed in this form, even if there are no n^2, n or constant terms (B, C and/or D = 0). For example, n^3 can be expressed as 1n^3 + 0n^2 + 0n + 0. We will confirm whether our solution has an n^2, n and constant term when we solve for B, C and D.
I don't understand why if the difference of the difference of the difference is constant I can express the sum of the series with a Third level polynomial :( and video that explains that?
Linear functions have a constant rate of change (slope). For example, f(x)=3x+5 changes by 3 for every single change in x. Thus, the difference in y value between any two consecutive x values is a constant 3. A quadratic function like g(x) = 2x^2 actually doesn't have a constant slope- its slope increases along the function, but its slope increases at a constant rate (calculus can show you that the slope of g(x), called g'(x), is equal to 4x). This means that if you were to calculate the difference in y values between any consecutive x values for g(x), that difference would increase by a constant amount (2, 6, 10, 14 etc), but the difference of the differences would be a constant 4. Now, a cubic function just extends this concept one step further: its slope not only increases, it increases at an increasing rate. That is, a cubic function's slope function is quadratic. Thus, you have to go three levels deep taking differences with a cubic function before you arrive at a constant rate of change. In general, the degree of the polynomial is the same as how many times you have to take the difference (of the difference of the difference...). This corresponds to taking the nth derivative in calculus. Hope that helps.
@@inordirection_ cool dude! I was just sitting here trying to think about how to prove this fact but with now success. Maybe one could use some combination of taking the n:th derivative and induction to prove it?
@@inordirection_ A 4 year old comment like this deserves a oscar for the incredible amount of information. Thank you so much! Btw If you do in fact see this, hope you are doing well :)
Thank you this is completely brilliant .... the only thing puzzling me is why you are so determinant on choosing the first polynomial equation .... This is the best explanation I have seen on this topic but still I can't see why it must be a polynomial of third degree. But still thank you for this video.
Because he found out that the difference of the difference of the difference is a constant, aka the third derivative of this function is a constant , so happens that the third derivative of a cubic function is a constant and that is exactly why it works here.
This proof is beautiful but not rigorous because you didn't proof that every difference of difference of difference is equal to 2. maybe should have done it by induction
ʃ֓֔œۡۓ He wrote the sum as a polynomial function , we can write those function like Ax^2+Bx+C which degree is 2 or Ax^4+Bx^3+Cx^2+Dx+E which degree is 4 etc. However I didn’t understand why he chose to use third degree also
@@ekinersozlu2305 because cubic function. 3rd degree polynomial form therefore appropriate. if the difference of the difference of the difference of the difference was constant, then it would be 4th degree polynomial, meaning the 4th derivative (not the third as would be the case here) is 0, meaning rate of change, meaning the difference is constant. connection to calculus is that speed is a rate of change in displacement, so it is change in x over change in time. if travelling at a constant speed, derivative is zero. however, when acceleration enters the picture, then you have rate of change of rate of change, and so on.
@maru as a tree Mr. khan is trying to encourage us to the elemental proof that could be easily understood if you had studied graph of parabola in calculus.
How can we say it is a 3rd order polynomial if it isn't defined for negative numbers? How can we say it is a 3rd order polynomial if the function is only defined for positive integers and zero? Aren't polynomials continuous and differentiable? The graph would have jump discontinuities.
John Bennett Yeah that’s what I was thinking . I don’t think it is a polynomial because, like you said, polynomials are continuous and this is not. But apparently, in this case, I guess they have similar behavior because of the way (rate) they progress from one to the next. I just wanted to put down my thoughts because you had this similar question as me 😂, I am not very clear on it yet, don’t take it as a valid explanation.
The difference of the difference of the difference was very useful, very cool how that works.
Hello
It's essentially taking the second derivative
this is by far the simplest explanaition i´ve found on the internet. actual ape likes your thingking methods.
@ Lud R - That is the general form of a cubic equation (An^3 + Bn^2 + Cn + D). Any cubic equation can be expressed in this form, even if there are no n^2, n or constant terms (B, C and/or D = 0). For example, n^3 can be expressed as 1n^3 + 0n^2 + 0n + 0. We will confirm whether our solution has an n^2, n and constant term when we solve for B, C and D.
Thanks for those videos, they help me to understand a lot about mathematics, thanks!
A=1/3
B=1/2
C=1/6
Thanks. This is what i was looking for. No one explained why cubic form is used to express square natural number sum.
Where can i learn more about the difference of the difference trick? Why does it work?
Wow, I never knew about that difference trick.
I don't understand why if the difference of the difference of the difference is constant I can express the sum of the series with a Third level polynomial :( and video that explains that?
Linear functions have a constant rate of change (slope). For example, f(x)=3x+5 changes by 3 for every single change in x. Thus, the difference in y value between any two consecutive x values is a constant 3.
A quadratic function like g(x) = 2x^2 actually doesn't have a constant slope- its slope increases along the function, but its slope increases at a constant rate (calculus can show you that the slope of g(x), called g'(x), is equal to 4x). This means that if you were to calculate the difference in y values between any consecutive x values for g(x), that difference would increase by a constant amount (2, 6, 10, 14 etc), but the difference of the differences would be a constant 4.
Now, a cubic function just extends this concept one step further: its slope not only increases, it increases at an increasing rate. That is, a cubic function's slope function is quadratic. Thus, you have to go three levels deep taking differences with a cubic function before you arrive at a constant rate of change. In general, the degree of the polynomial is the same as how many times you have to take the difference (of the difference of the difference...). This corresponds to taking the nth derivative in calculus. Hope that helps.
@@inordirection_ cool dude! I was just sitting here trying to think about how to prove this fact but with now success. Maybe one could use some combination of taking the n:th derivative and induction to prove it?
@@inordirection_ Bruh, holy cow this is sick. Thanks man :)
This community is insane
@@inordirection_ Greatly helpful; thank you 💯
@@inordirection_ A 4 year old comment like this deserves a oscar for the incredible amount of information. Thank you so much! Btw If you do in fact see this, hope you are doing well :)
Thank you this is completely brilliant .... the only thing puzzling me is why you are so determinant on choosing the first polynomial equation .... This is the best explanation I have seen on this topic
but still I can't see why it must be a polynomial of third degree. But still thank you for this video.
I wish somebody could explain. It's funny how almost every time there's a different step in a problem, it's the only part the video doesn't explain
@@sazob 😆
Because he found out that the difference of the difference of the difference is a constant, aka the third derivative of this function is a constant , so happens that the third derivative of a cubic function is a constant and that is exactly why it works here.
beautiful method
FINALLY THANK YOU
This proof is beautiful but not rigorous because you didn't proof that every difference of difference of difference is equal to 2. maybe should have done it by induction
A= 1/3. B=1/2, C=1/6
A=1/3, B = 1/2 and C = 1/6
math is beautiful
Jesus how did you get An"'+Bn"Cn+D
ʃ֓֔œۡۓ He wrote the sum as a polynomial function , we can write those function like Ax^2+Bx+C which degree is 2 or Ax^4+Bx^3+Cx^2+Dx+E which degree is 4 etc. However I didn’t understand why he chose to use third degree also
@@ekinersozlu2305 because cubic function. 3rd degree polynomial form therefore appropriate. if the difference of the difference of the difference of the difference was constant, then it would be 4th degree polynomial, meaning the 4th derivative (not the third as would be the case here) is 0, meaning rate of change, meaning the difference is constant. connection to calculus is that speed is a rate of change in displacement, so it is change in x over change in time. if travelling at a constant speed, derivative is zero. however, when acceleration enters the picture, then you have rate of change of rate of change, and so on.
WHAT JUST BECAUSE DIFFERENCE OF DIFFERENCE OS THE SAME DOES NOT.MEAN AMYONE WILL THINK OF CUBIC FUNCTIONS OR IT IS RELATED AT ALL TO CUBJCS!!
Super
I heard Euler found a way to show that this equation when n is infinity is zero, but I can't find how he did it, anyone knows?
It's not the same thing as this, but it's fairly similar. This should help you: en.wikipedia.org/wiki/E_(mathematical_constant)
Does anyone know the sequence to
n: 1 2 3
A(n): 6 12 20
@@joemorrison4143 f(n) = n^2+3n+2, so the next number is 30. Hope it helped :p
Where does Sal get those N^3 .... N^2 and N from!?
I hope you found the answer to your question by now.
@moyenmedium1220 ots still a DIRTY CHEAT..THIS IS CHEATING..NONONE IS GOING TO THNK OF CUBED OUT OF NOWHERE LIKE THAT..
Cramer :)
There are better proofs out there. This one I don't like.
@maru as a tree Mr. khan is trying to encourage us to the elemental proof that could be easily understood if you had studied graph of parabola in calculus.
Isn't there ANY ACTUAL PROPF NOT INVOLVING CUBE FORNULAS AT ALL..JUST ALGEBRA FOR HEAVENS SAKE!!
this doesnt have a begining and it is really confusing
How can we say it is a 3rd order polynomial if it isn't defined for negative numbers?
How can we say it is a 3rd order polynomial if the function is only defined for positive integers and zero?
Aren't polynomials continuous and differentiable? The graph would have jump discontinuities.
John Bennett
Yeah that’s what I was thinking . I don’t think it is a polynomial because, like you said, polynomials are continuous and this is not. But apparently, in this case, I guess they have similar behavior because of the way (rate) they progress from one to the next. I just wanted to put down my thoughts because you had this similar question as me 😂, I am not very clear on it yet, don’t take it as a valid explanation.
@@someonesomeone4099 can't we describe it as a sequence ?
When you’re watching this already knowing calculus and know that an nth degree function’s nth derivative is always a constant.
But i is used for imaginary numbers lol
i dont really understand the difference at the start of the video at abt 3 mins