system
Given the following post, respond as if you're commenting on Reddit:
human
Is there a reason for the difference between cubes (numbers to the power of 3)?

If you take the first six cubes:

1   8   27   64   125   216

Then calculate the differences:

7   19   37   61   91

Then the difference between those numbers:

12   18   24   30

The difference between these numbers is always 6. Is there an explanation as to why this is?
gpt
So, you have four consecutive cubes, let's say n^(3), (n+1)^(3), (n+2)^(3), and (n+3)^(3).

The differences between the successive cubes are (n+1)^(3) \- n^(3), (n+2)^(3) \- (n+1)^(3), and (n+3)^(3) \- (n+2)^(3).

The differences between the differences are ((n+2)^(3) \- (n+1)^(3)) - ((n+1)^(3) \- n^(3)) and ((n+3)^(3) \- (n+2)^(3)) - ((n+2)^(3) \- (n+1)^(3)).

And finally the difference between the last two numbers is (((n+3)^(3) \- (n+2)^(3)) - ((n+2)^(3) \- (n+1)^(3))) - (((n+2)^(3) \- (n+1)^(3)) - ((n+1)^(3) \- n^(3))). Simplify this expression. See what you get.

Basically, the first time you take the difference, the cubes cancel out, next time the squares cancel out, and the third time the linear terms cancel out leaving only a constant.

Btw, this phenomenon is intimately tied with differentiation. The third derivative of the third-degree polynomial is a constant function precisely because of the behavior you noticed.