Show that all Polygonal sequences generated by the Series of Arithmetic sequence with common difference #d, d in ZZ# are polygonal sequences that can be generated by #a_n = an^2+bn+c#?
example given an Arithmetic sequence skip counting by
you will have a
A polygonal sequence is constructed by taking the
So the key hypothesis here is:
Since the arithmetic sequence is linear (think linear equation) then integrating the linear sequence will result in a polynomial sequence of degree 2.
Now to show this the case
Start with a natural sequence (skip counting by starting with 1)
find the nth sum of
So with d = 1 the sequence is of the form
Now generalize for an arbitrary skip counter
Which is a general form