Let q be the amount of bacteria after n minutes. We know that the amount of bateria triples every 20 seconds, this means that in one minute the amount of bacteria will triples 3 times, that is the amount triples every:
[tex]\frac{1}{3\text{ }}\text{ min}[/tex]The expression we are looking for has to be of the form:
[tex]q(n)=Q(3^{kn})[/tex]So we need to find the coefficient k. To do this we know that after 1/3 of a minute the amount will be 3Q, then we have:
[tex]\begin{gathered} q(1)=Q(3^{\frac{1}{3}k}) \\ 3Q=Q(3^{\frac{1}{3}k}) \\ 3=3^{\frac{1}{3}k} \\ \text{which implies that:} \\ \frac{1}{3}k=1 \\ k=3 \end{gathered}[/tex]Therefore the equation we are lokking for is:
[tex]q(n)=Q^{}(3^{3n})[/tex]