Nonsensical question on recurrence relations. Can it be fixed? Problem. A biologist takes $3$ cells and observes them in his lab as they duplicate themselves.  Each step of time he observes that the new cells that are "born" is the double of the number of cells in the previous step, but a third of the number of cells of the previous step actually die.  Write a recurrence relation for the number of cells that are alive at time step $n$.
Solution. Straight from the problem, we know
$$c_n = \begin{cases}3 & \text{if } n = 0\\
                   c_{n - 1} + 2c_{n -1} - c_{n - 1}/3 & \text{if } n > 0.
\end{cases}$$
So far, so good.  But the problem is that $c_1 = 8$ and $c_2 = 8 + 16 - 8/3 = 24 - 8/3$.  In other words, at step two, we need to kill six cells and then kill $2/3$ of another.  How am I going to kill $2/3$ of a cell?
Is it possible to ignore this imperfection?  Can I replace "cell" with something that's "always" (if you will) divisible, like ounces of water?

Allow fractions of cells. After very few time steps the number will be large enough so that the fractions will be insignificant.
This is a toy problem that's poorly written. The initial condition makes little biological sense. And "time instant" should be "time step".
