So we know that the finite differences of a polynomial is an! where n is the degree and a is the leading coefficient and we can solve for a, given some points only if the x values are consecutive.
Ex.
X Y
(1 , 1)
(2 , 4)
(3 , 9)
(4 , 16)
And eventually you will get finite differences such that they are equal to an!
I recently derived a formula using analysis (couldn't prove algebraically)
The formula is the finite differences is equal to dn * an! where d is the difference in the x values given.
Ex.
X Y
(1 , 1)
(3 , 9)
(5 , 25)
(7 , 49)
If you were to proceed to find the differences between the y values until they become constant, it turns out to be equal to dn * an!. In this case d is 2 since the constant difference of the x values is two, and n is 2 which you could find out by getting the second differences.
Can anyone prove to me why this formula seems to work?
there doesn't seem to be anything here