Since all of the differences are the same after two passes, you know a quadratic will suffice for a rule (one pass: linear, three passes: cubic, etc.)
So you need a function of the form
Plugging in numbers you get
Which leads to the following 3X3 system:
Which you can solve by your favorite process to get the coefficients of your function.
There is actually a shortcut for sequences. The lead coefficient of your function is always one-half of the value of the value of the constant difference. The constant difference here was 1, so the lead coefficient is 1/2.
John
My calculator said it, I believe it, that settles it
