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I'm familiar with exponential growth, however I'm not sure how deal with situations where my quantities are discrete and rounding errors come into play.

To be concrete, say I got $N$ items arranged in a line. Now I have every fourth of them replaced by two items. So approximately, we've got $N' = N + \frac N 4 = \frac 5 4 N$ items in the first step. Repeating the process, we have exponential growth.

Precisely however, we need to take into account that our $N$ usually cannot be split evenly into blocks of four, so we've got like

xxxxxxxxx (N=9)

xxx xx xxx xx x (N'=9+2)

So our correct iteration formula would be $$N_{n+1} = N_n + \Big \lfloor \frac N 4 \Big \rfloor$$ Now is there a closed form solution for $N_n$ after $n$ iterations, much like the exponential growth formula in the continuous case? Can one still apply some growth factor $\frac 5 4$ anywhere?

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