Question

Suppose that the number of cells in a tumor doubles every 5 months. Considering that the tumor begins with a single cell, answer the following: How many cells will there be after 3 years? 5 years?

Answer 1

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.