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