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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

Respuesta :

let [tex]a_n[/tex] represent the number of bacteria after [tex] \frac{n}{2} [/tex] hours.

so

[tex]a_0[/tex] is the number of bacteria after 0/2 = 0 hours

[tex]a_1[/tex] is the number of bacteria after 1/2 hours

[tex]a_2[/tex] 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:

[tex]a_0=1[/tex]

[tex]a_1=1*2[/tex]

[tex]a_2=1*2*2[/tex]

[tex]a_3=1*2*2*2[/tex]

[tex]a_4=1*2*2*2*2[/tex]


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

 [tex]a_4_8[/tex] is the number of bacteria after
 
[tex]\frac{n}{2}=\frac{48}{2}= 24[/tex] hours.

Thus, we calculate  [tex]a_4_8[/tex], which is [tex] 2^{48} [/tex].


Answer: the number of bacteria after 24 hours is [tex] 2^{48} [/tex].

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