Question: Suppose that there are two goats on an island initially. The number of goats on the island doubles every year by natural reproduction, and some goats are either added or removed each year.
a) Construct a recurrence relation for the number of goats on the island at the start of the nth year, assuming that during each year an extra 100 goats are put on the island.
b) Solve the recurrence relation from part (a) to find the number of goats on the island at the start of the nth year.
c) Construct a recurrence relation for the number of goats on the island at the start of the nth year, assuming that n goats are removed during the nth year for each n ≥ 3.
d) Solve the recurrence relation in part (c) for the number of goats on the island at the start of the nth year.