Shortcut of finding the explicit formula for a quadratic sequence (with the proof, too!)

[@msolec2000]

IIRC, Mathologer gave the method for any polynomial sequence, if you start with n=0, but easy change afterwards... you would go, in your examples (nC0)·0 + (nC1)·4 + (nC2)·2 (That is n choose 0, n choose 1, so on), and the second example (nC0)·(-2) + (nC1)·5 + (nC2)·4, use always the leftmost results of each line.

[@MarioFanGamer659]

For context, that's "Why don't they teach Newton's calculus of 'What comes next?'".

[@General12th]

What an amazing technique! (And proof!)

[@BigRedDino3311]

2 months of school and I didn't get it. 1:28 into the video and I get it. Thanks, American school system.

[@Brid727]

this method is so good omg, everytime id get a quadratic sequence id also put 2A equal to the difference of the sequence of difference of the original sequence, but after that id print a sequence using An^2, subtract that from the original sequence, get another arithmetic sequence(must work if you do it correctly), find the nth term of that sequence(the Bn + C part) and pair everything to get the nth term = An^2 - Bn + C i hated to do that much work and this video helped me to save my sanity and time

[@vaishnavisingh6654]

Sir can't you give any shirt trick for calculating cube roots and higher roots ?

[@jamescollier3]

Haha. That was a very quick exi

[@kennethalbert5903]

am i dumb or is there a mistake on the derivative part? I think i understand what u mean by taking the derivative (is to find the equation of the rate of the series right?) but for the last question where the answer is (-2,3,12,25,45) 2n²-n-3 but if i take the derivative its 4n-1 however the difference series is supposely 5,9,13,17 (but this is 4n+1?) not 4n-1, am I missing something?

[@farhansadik5423]

Thanks for this comment man, At first when I read this comment i thought this guy must've made a dumb mistake somewhere, after a day's worth of searching and studying i'm now in the same mess, and finally understand where the discrepancy comes from. Basically, look at the function that he created. It's a quadratic function, in which the graph is continous for all x in it's domain. However, we are restricting the domain by only allowing x/n to be positive integer values for us to get the sequnce. The discrepancg arises from the fact the derivatives work on Infinitesimal small steps on the x axis and measuring the change in y. But as i said we restricted the x to only change with steps of 1 because we needed the sequence. In other words the concept of derivative doesnt apply perfectly to this because 1 is most definitely not a infinitely small step :) so the actual difference given by the sequence diverges from the true difference which would have been given by the quadratic. Lemme clear it again. Suppose you sequenced the values of 2x^2-1 for ALL X Meaning for all and each x (literally infinite values) you wrote out the values. If you( a superman) did that, and then calculated the difference between the values, they would be exactly the derivative of 2x^2-1 ((4x)). Please reply again if you have any questions, i will glady answer again since i learned so much

[@kennethalbert5903]

@@farhansadik5423 hi dude thanks for the reply I actually realise the issue of this method, suppose we have 2,4,8,14 The 2nd derivative of this is 2, always the 1st derivative is 2x+c, in this case notice how the pattern goes to 2,4,6(but this is actually in the middle of it) so f'(1.5) = 2, f'(2.5) = 4, so on.. using f'(1.5) = 2, we can find that the c is -1 so its 2x-1, (1st derivative), so the original is x²-x+c, now we know f(1) = 2, and f(2) = 4, f(3) = 8, f(4) = 14, using any of these... f(1) = 2, gives us that the c (for original) is 2, so f(x) = x²-x+2, its quite beautiful actually that this "middle" thing is what makes the difference. Also thanks again for replying despite it being 5 months ago lol.

[@farhansadik5423]

@@kennethalbert5903 yep! That's in part with what is the concept i wrote. this was a nice thing since i learned a lot thanks again, have a good day brother

[@jcrodummy]

Thanks

[@leonardobarrera2816]

I didn't understand the second step to find b

[@davidhowe6905]

He found A=1 so 3A=3, then subtracted 3 from both sides. He mentioned this verbally but it was easy to miss in the first example.

[@caniraso]

You can do that 3rd degree equation also. This time 6A=d. 4th degree 24A=d 5th degree 120A=d you see the pattern.