2016 DHS P2 Q3

The number of bacteria (in millions) in Pond \({A}\) at the start of the \({n}\)th week, before any chemical treatment, is given by \({u_n}\). Pond \({A}\) is treated at the start of each week with Chemical \({A}\), which kills \(70\%\) of all bacteria instantly. At the end of each week, \(6\) million new bacteria is reproduced.

1. Write down a recurrence relation of the form \({u_{n+1} = au_n + b}\), where \({a}\) and \({b}\) are constants to be determined.[1]
2. Show that \({{u}_{n}}={{0.3}^{n-1}}{{u}_{1}}+\frac{60}{7}\left( 1-{{0.3}^{n-1}} \right)\).[2]

The number of bacteria (in millions) in Pond \({B}\) at the start of the \({n}\)th week, before any chemical treatment, is given by \({v_n}\).

Pond \({B}\) is treated at the start of each week with Chemical \({B}\). It is known that \({v_n}\) follows the recurrence relation

\({{v}_{n+1}}=0.01{{v}_{n}}^{2}+6\)

It is given that if the sequence \({{v}_{1}}\), \({{v}_{2}}\), \({{v}_{3}}\), … converges to a limit, it converges to either \({\alpha}\) or \({\beta}\), where \({\alpha < \beta}\).

3. Find \({\alpha}\) and \({\beta}\). Explain whether \({v_n}\) necessarily converges to \({\alpha}\) or \({\beta}\).[3]
4. If \({u_1 = v_1 = 30}\), determine which chemical would be more effective in killing the bacteria in the long run.[2]

Pond \({C}\) is treated with Chemical \({C}\). To account for the bacteria’s increasing resistance to the chemical, the dosage of Chemical \({C}\) is increased by \(5\) ml each week. The first dose is \(20\) ml.

5. How many weeks does it take to finish the first \(3\) litres of chemical in the treatment of Pond \({C}\)?[3]