What does the common difference of a sequence describe
I am given the sequence 1, 7, 21, 43, 73, ... and can derive without any assumption about the nature of the sequence that the ultimate difference is 8, I'm interested in understanding what this difference allows you to deduce about the original polynomial which generates this sequence; and whether it is possible to find this polynomial?
I can see that 1, 7, 21, 43, 73, ... has a difference of 6, 14, 22, 30, ... which is generated by the arithmetic sequence 6 + 8(n-1), but trying to rationalize how the difference of a sequence can then be non-constant (1, 7, 21, 43, 73) it doesn't really make sense.
Can anyone shed some light on how to go about solving this intuitively?