Answer:
Bacteria replicate by binary fission, a process by which one bacterium splits into two. Therefore, bacteria increase their numbers by geometric progression whereby their population doubles every generation time.
The general formula for a geometric sequence is[tex]\[{{a}_{n}}={{a}_{1}}{{r}^{n-1}}\][/tex]
where [tex]\[{{a}_{1}}\] = first term[/tex] and[tex]\[{{r}_{1}}\]= common ratio[/tex].
Step-by-step explanation:
• Given [tex]\[{{a}_{1}}=4\][/tex] and [tex]\[{{a}_{5}}=324\][/tex] , we have to find [tex]\[{{a}_{2,}}{{a}_{3}}\][/tex] and [tex]\[{{a}_{4}}\][/tex].
• In order to find the common ration consider about formula [tex]\[{{a}_{n}}={{a}_{1}}{{r}^{n-1}}\][/tex]
[tex]& 324=4{{(r)}^{5-1}} \\ & 81={{(r)}^{4}} \\ & {{(3)}^{4}}={{(r)}^{4}} \\ & r=3 \\ \end{align}[/tex]
Bacteria found after 2 hour is
[tex]& {{a}_{2}}=4{{(3)}^{2-1}} \\ & {{a}_{2}}=12 \\ \end{align}[/tex]
Bacteria found after 3 hour is -
[tex]& {{a}_{3}}=4{{(3)}^{2}} \\ & {{a}_{3}}=36 \\ \end{align}[/tex]
Bacteria found after 4 hour is –
[tex]& {{a}_{4}}=4{{(3)}^{4-1}} \\ & {{a}_{4}}=4(27)=108 \\ \end{align}[/tex]
• Hence the bacteria found at 2,3 and 4 hours is 12,36 and 108.
Learn more about bacteria growth here:
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