6 When \(10 ^ { 6 }\) of a certain type of bacteria are detected in a blood sample of an infected animal, a course of treatment is started. The long-term aim of the treatment is to reduce the number of bacteria in such a sample to under 10000 . At this level the animal's immune system can fight the infection for itself. Once treatment has started, if the number of bacteria in a sample is 10000 or more, then treatment either continues or restarts.
The model suggested to predict the progress of the course of treatment is based on the recurrence system \(P _ { n + 1 } = \frac { 2 P _ { n } } { n + 1 } + \frac { n } { P _ { n } }\) for \(n \geqslant 0\), with \(P _ { 0 } = 1000\), where \(P _ { n }\) denotes the number of bacteria (in thousands) present in the animal's body \(n\) days after the treatment was started.
The table below shows the values of \(P _ { n }\), for certain chosen values of \(n\). Each value has been given correct to 2 decimal places (where appropriate).
| \(n\) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
| \(P _ { n }\) | 1000 | 2000 | 2000 | 1333.33 | 666.67 | 266.67 | | 25.47 | 6.64 | 2.68 |
| \(n\) | 10 | 20 | 40 | 60 | 80 | 100 | 200 | 300 | 400 |
| \(P _ { n }\) | 3.89 | 4.67 | 6.45 | 7.84 | 9.03 | 10.08 | 14.20 | 17.36 | 20.04 |
- Find the value of \(P _ { 6 }\) correct to 2 decimal places.
- Using the given values for \(P _ { 0 }\) to \(P _ { 9 }\), and assuming that the model is valid,
- describe the effects of this course of treatment during the first 9 days,
- state the number of days after treatment is started when the animal's own immune system is expected to be able to fight the infection for itself.
- Using information from the above table, suggest a function f such that, for \(n > 10 , \mathrm { f } ( n )\) is a suitable approximation for \(P _ { n }\).
- Use your suggested function to estimate the number of days after treatment is started when the animal may once again require medical intervention in order to help fight off this bacterial infection.
- Using information from the above table and the recurrence relation, verify or correct the estimate which you found in part (c)(ii).
- One criticism of the system \(P _ { n + 1 } = \frac { 2 P _ { n } } { n + 1 } + \frac { n } { P _ { n } }\), with \(P _ { 0 } = 1000\), is that it gives non-integer
values of \(P\). values of \(P _ { n }\).
Suggest a modification that would correct this issue.