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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.

  1. Construct a recurrence relation for the number of goats on the island at the start of the $n^{\text{th}}$ year, assuming that during each year an extra $100$ goats are put on the island.
  2. Solve the recurrence relation from part $(A)$ to find the number of goats on the island at the start of the $n^{th}$ year.
  3. Construct a recurrence relation for the number of goats on the island at the start of the $n^{\text{th}}$ year, assuming that n goats are removed during the $n^{\text{th}}$ year for each $n \geq 3.$
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