Hi celestia007,

I guess you are asking about "Arithmetic progressions" and the two formulas involved in your posts are (seems to me) related to properties of TERMS and of SUM OF TERMS.

When we write An = A1 + (n-1).d we mean that the nth term (say a5) is equal to the first term (a1) plus 4 times the difference between any two consecutive terms (d), in this case we would have: a5 = a1 + 4.d

Example: arithmetic sequence given by: 4, 7, 10, 13, 16, ... please note that a1 = 4, a5 = 16 and d = 3 (7-4 = 10-7 = ...)
and the formula is right: a5 = a1 + 4.d (check it).

When we write Sn = S(n-1) + An we mean that the Sum of the terms from the first to the nth is equal to the sum of the terms from the first till the (n-1)th plus the An th term. In our case, please note that S4 = 4+7+10+13 equals S3 = 4+7+10 plus A4 = 13 etc etc.

Now you must see the Manhattan Prep´s guide carefully to understand what Kn, K and X are at that context... because from what I wrote to you I guess K would be the An th term (from what you called Recursive formula) and then the first formula you mentioned Sn = Kn + X would not (seems to) be related accordingly...

As far as which formula is better, each one has its own value and usage (as usual), so you should (my opinion) start understanding "the players" before trying to choose "how to play the game".

I hope I was of some help.

Regards,
Fabio.