12. Initially the number of fish in a lake is 500000 . The population is then modelled by the recurrence relation
$$u _ { n + 1 } = 1.05 u _ { n } - d , \quad u _ { 0 } = 500000 .$$
In this relation \(u _ { n }\) is the number of fish in the lake after \(n\) years and \(d\) is the number of fish which are caught each year.
Given that \(d = 15000\),
- calculate \(u _ { 1 } , u _ { 2 }\) and \(u _ { 3 }\) and comment briefly on your results.
Given that \(d = 100000\),
- show that the population of fish dies out during the sixth year.
- Find the value of \(d\) which would leave the population each year unchanged.