Question

Suppose that a colony of bacteria starts with 1 bacterium and doubles in number every half hour. how many bacteria will the colony contain at the end of 24

Answer 1

Let $a_n$ represent the number of bacteria after $\frac{n}{2}$ hours.

so

$a_0$ is the number of bacteria after 0/2 = 0 hours

$a_1$ is the number of bacteria after 1/2 hours

$a_2$ is the number of bacteria after 2/2 = 1 hours and so on


so if we list a few terms of this sequence, we have:

$a_0=1$

$a_1=1*2$

$a_2=1*2*2$

$a_3=1*2*2*2$

$a_4=1*2*2*2*2$


so clearly, $a_n$, the number of bacteria after $\frac{n}{2}$ hours, is equal to $2^{n}$.
 

 $a_4_8$ is the number of bacteria after
 
$\frac{n}{2}=\frac{48}{2}= 24$ hours.

Thus, we calculate  $a_4_8$, which is $2^{48}$.


Answer: the number of bacteria after 24 hours is $2^{48}$.