# 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#?

##### 1 Answer

#### Answer:

with

example given an Arithmetic sequence skip counting by

you will have a

#### Explanation:

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

with

Now generalize for an arbitrary skip counter

Which is a general form

with