# Explain the next number in these sequences

For the purposes of my work, I have been researching upgrade costs in a video game. Each sequence of numbers below relates to a different researchable element. The numbers per sequence show the rise in cost each time the previous value is paid by the player.

Any help in figuring out how these different sequences grow would be greatly appreciated.

0.50, 3.59, 7.61, 13.12, 20.27, 30.00, 42.66 ......?

2.15, 20.82, 48.33, 88.34, 147, 234, 360 ......?

1.06, 187, 1239, 6906, 36461, 187096 ......?

• there is a question in math.stackexchange with the same values but no answer yet math.stackexchange.com/questions/1883198/…. unless you are aTop – lois6b Aug 5 '16 at 9:53
• Is that last 187,096 meant to be 187.096 or 187096 or 187, 96? (Same typo[?] appears at math.stackexchange.) – YowE3K Aug 5 '16 at 10:06
• being 096 i think is a typo... should be a "." – lois6b Aug 5 '16 at 10:09
• Have you confirmed that the sequences are repeatable (if you replay the game) - or may there be a stochastic element? – Jonathan Allan Aug 5 '16 at 17:32

I realize this isn't a math forum and that it's puzzling (it's not a pun), but I have these most reasonable answers by mathematical methods (regressions): (I'll also put in less math-y definitions)

All three equations appear to be power functions, or those of the form $y=ax^k$, where $y$ is the upgrade cost, $a$ is a constant, $x$ is the number of upgrades bought, and $k$ is another constant. $y = ak^x$ is another close fit, or an exponential function, following variable definitions above.

Note: correlation coefficient represents how close a regression line/curve (line/curve of best fit) is; $R^2$ close to 0 means almost no fit (or points everywhere) and $R^2$ close to 1 is really good fit, or points are really close to the line of best fit.

First equation: $y = 2.0858x^{1.6732}$, correlation coefficient $R^2 = 0.996$
Another good fit: $y = 3.464(1.5265)^x$, correlation coefficient $R^2 = 0.987$

Second equation: $y = 8.9852x^{2.0488}$, correlation coefficient $R^2 = 0.995$
Another good fit: $y = 19.943(1.6244)^x$, correlation coefficient $R^2 = 0.993$

Third equation: $y = 1.6153x^{7.2444}$, correlation coefficient $R^2 = 1$
Another good fit: $y = 51.982(5.1434)^x$, correlation coefficient $R^2 = 1$

Except the first values they all seem to grow exponentially, at least in an asymptotic meaning. That is, the ratio of consecutive elements is almost constant. You can give some guesses for the next terms by multiplying the last term in the sequences by the powers of the constant which you can estimate that way.