# What general function or rounding method can be derived from these number series?

OK, first of all, don't laugh, this is related to a social iOS game, but I promise there is some real juicy math here...

I am attempting to derive a general formula or algorithm that can predict the next 9 numbers in a series based on a single starting number. Every series has exactly 10 numbers, no more, no less. Here are four samples that use the correct (unknown) algorithm:

 Starting numbers: ("Initial cost")
7,500       4,000       2,500       1,000

Next 9 numbers: ("Upgrades")
12,500      6,680       4,175       1,670
21,000      11,000      7,000       2,790
35,000      18,500      11,500      4,660
58,000      31,000      19,000      7,775
97,500      52,000      32,500      13,000
163,000     87,000      54,000      21,700
270,000     144,000     90,000      36,250
450,000     240,000     150,000     60,500
750,000     400,000     250,000     101,000


More examples can be found at the following forum page under "money buildings", ignoring "gold buildings". (Remember, I said don't laugh.) The data here is good; I have confirmed both the validity of the sample data provided, and that these series are indeed based on only the "initial cost" by inspecting the manifest files that are sent by the game manufacturer (they only contain the first number in the series, implying that the game's code figures out the rest).

Summary of my findings so far:

• Each number is approximately 5/3 of the previous number, but not exactly.
• There is no obvious pattern surrounding the rounding up or down of values.
• Extrapolating a simple exponential trend (in Excel) does not produce an exact or "close enough" solution.
• I have solved other algorithms based on other series in this game, and some involved rules, such as "the 10th item in the list shall be multiplied by constant X", so it is possible that similar rules exist.
• There is a small chance that the numbers in this series is dependent on other series not listed here but is listed on the forum page linked above. If someone makes a good case for it, I can include that data here as well.
• My best guess for the actual solution probably involves some quirky rounding method.

It is worth mentioning that I have already attempted using the OEIS suggested by the excellent question, "Predict next number from a series". Along with my own trials and errors, I have hit a dead end and am hoping that someone in the community here can figure this out.

I will accept any of the three following answers:

1. Proof that an algorithm cannot exist or cannot be found without psychic powers.
2. An exact answer.
3. A general approach that will eventually lead to an answer.

(If the solution turns out to be interesting, it would be a good candidate to submit to the OEIS.) And for those wondering why, there is no need for a "why", I am just trying to learn new techniques for solving these types of problems, if they exist.

-
Your question can be read in two ways. Do you mean "I have these lists of starting numbers and I want to predict the next 9" or do you mean "I am looking for a general way to predict the next 9 terms in a sequence from the first which applies to all sequences"? (The latter is obviously impossible.) –  Qiaochu Yuan Jun 1 '12 at 14:40
Hi Qiaochu, thanks for the comment. Perhaps I can rephrase the question. I would like to predict the next 9 numbers in a series given only the first number. Only the first number is provided within the game's code, and the other 9 must be derived mathematically. The sample data I have above was determined experimentally. –  Kevin McCormick Jun 1 '12 at 14:48
Ah, sorry, I misread your question initially. Okay, now it's a little clearer that you only want a specific answer for this particular game. –  Qiaochu Yuan Jun 1 '12 at 14:52
Yes, this answer does apply directly to this game, however I am hoping a that a general approach may exist to solving these types of problems, or that the solution to this specific problem will be interesting/helpful when solving other similar problems. –  Kevin McCormick Jun 1 '12 at 14:55
Looking at these numbers, i would say that a sequence of the type $x_n=x_0a^n$ should work fairly well. At each step it almost doubles, so $a$ should not be very far from 2. But that is an approximation, nothing exact. –  guaraqe Jun 1 '12 at 14:57