# Demystify / Solve a number progression.

I've worked on this for two days and haven't gotten anywhere.

They don't seem to grow by an even percentage, nor by an incremented percentage, nor by a flat number increment. (as far as my limited math know-how goes)

Can anyone figure it out?

$$240, 470, 904, 1701, 3138, 5672, 10049$$

If you know how to do it, please explain, rather than just giving me the formula.

As for context, its the rising price of a consumable in a custom board game we play every once in a while, I'm just trying to figure out if the guy who oversees the rules is pulling the numbers out of his imagination or he actually has a plan, as he doesn't miss out on reminding us, he's better at planning than we are ...
More context ... What i do for a living is mostly creative writing and some design (pretty much squeezed out of templates from whatever software I end up using). So advanced math terms and explanations are just beyond me. Not to mention that I tackle this when the others go on smoke breaks and after work when I'm tired. Before last month I didn't know exactly how to find what % A is of B. For those helping - please try and do it in simple common terms ... Also i suspect the solution is somewhat simpler. It is a board game. True the man who runs it has SOME math knowledge, and i suspect uses some sort of increments in his work, but i doubt anybody would go to great mathematical length to make the prices of the utilities in a personal game to play with friends on the weekends.

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## 1 Answer

Look for Lagrange Interpolation Formula. It guarantees any finite sequence of numbers has a pattern: that is, there is a formula that produces your sequence $a_0,a_1,a_2,\ldots, a_n$ as values of a polynomial at $0,1,2\ldots, n$. It will be a polynomial of degree at most $n$.

Given two numbers $a_0, a_1$ this reduces to showing that there is a line in the plane passing through $(0,a_0)$ and $(1, a_1)$. For three numbers there is a parabola (or in the degenerate case a line) passing through $(0,a_0), (1,a_1), (2,a_2)$. And so on.

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@P Vanchinathan ... I got a neat curve out of it, didn't get far with interpreting it though ... tinyurl.com/m8ufqo5 – Helen May 7 '14 at 9:55
Of course there is nothing sacred about the solution provided by Lagrange formula: for a sequence like $-1,1,-1,1,-1$ common sense might say next term is $+1$. Bu this formula fitting a 4th degree curve will give something different. The point I wanted to make was any sequence is generatable as polynomial values at equidistant points. – P Vanchinathan May 7 '14 at 10:34