recurrence relation number of bacteria

Question

Assume that growth in a bacterial population has the following properties:

1. At the beginning of every hour, two new bacteria are formed for each bacteria that lived in the previous hour.
2. During the hour, all bacteria that have lived for two hours die.
3. At the beginning of the first hour, the population consists of 100 bacteria.

Derive a recurrence relation for the number of bacteria.

I know that if the bacteria wouldn't die $A_{n} = 2A_{n - 1}$, but now I have no idea what to do.

Answer 1

We know $A_1 = 100$, and in the second hour there will be $200$ newborn and $100$ that will die the third hour. So, $A_2 = 300$. Those $300$ alive in the second hour mean that in the third hour there will be $600$ newborn and $200$ still alive; $A_3 = 800$.

Following this logic, we can define: $$A_n = 2 \, A_{n-1} + 2 \, A_{n-2}.$$

Answer 2

At time $n$:

• The bacteria that have been there at the previous time (time $n-1$) give rise to 2 bacteria each: $A_{n-1} \to 2A_{n-1}$

• The bacteria that were already there at time $n-2$ die: $A_{n-2} \to 0$

Put together, this means you get $$ A_n = 2 A_{n-1} - A_{n-2}. $$