Let's call a 5-month period a "pent", so the conversion from years to pents is 1 year for every 2.2 pents.
Then
[tex]3\text{ years}\times\dfrac{2.2\text{ pents}}{1\text{ year}}=6.6\text{ pents}[/tex]
and
[tex]5\text{ years}\times\dfrac{2.2\text{ pents}}{1\text{ year}}=11\text{ pents}[/tex]
Now, if the number of cells doubles every pent, and if [tex]C_n[/tex] denotes the number of cells after [tex]n[/tex] pents, then the number of cells is modeled recursively by
[tex]C_{n+1}=2C_n[/tex]
starting with [tex]C_0=1[/tex].
Solving explicitly for [tex]C_n[/tex], you arrive at
[tex]C_{n+1}=2C_n=2^2C_{n-1}=2^3C_{n-2}=\cdots=2^{n+1}C_0[/tex]
or,
[tex]C_n=2^nC_0=2^n[/tex]
So, after 6.6 pents (3 years), you should expect the number of cells to grow to about [tex]2^{6.6}\approx97[/tex]. After 11 pents (5 years), you should find [tex]2^{11}=2048[/tex] cells.
