Number Patterns

A botanist is monitoring the growth of a certain plant. She recorded the number of leaves and flowers of the plant with respect to time, which is shown in the following tables.

1. Complete the first table that records the number of leaves and the number of flowers of this plant for the first six weeks.
   
2. Based on your answers in (i), find a formula for the number of leaves in the \(k^{th}\) week in terms of \(k\), where \(1 \le k \le 10\).
3. Hence, find the number of leaves of this plant in week \(8\).

From the \(11^{th}\) week, the leaves of the plant start falling, meanwhile the plant starts blooming. Assume the plant's leaf-falling and flower-blooming are at two fixed rates, respectively. On the \(m^{th}\) week, all the leaves have fallen.

4. Complete the second table that records the number of leaves and the number of flowers of this plant since the \(11^{th}\) week.
   
5. Based on your answers in (iv), find a formula for the number of flowers in the \(n^{th}\) week in terms of \(n\), where \(n \ge 11\) (i.e. \(n\) exceeds \(10\)).
6. Find the values of \(m\) and \(p\).

Source clarification: the first rule gives 99 leaves at week 10, while the second table starts at 120 leaves in week 11. Treat week 11 as the starting observation for the later model; the tabulated fall of 10 leaves per week begins after that observation.