Question $1$: $(10$ points$)$

A fish farmer has $20,000$ salmon in her farm. The number of salmon increases by $5\%$ every month, and the farmer harvests $2000$ salmon each month.

$($a$)(4$ points$)$ Write a recursive sequence $P_n$ for the salmon population after $n$ months. Explain the meaning of each of the terms of the equations for $P_n$ and $P_0$ in words.

$($b$)(3$ points$)$ How many salmon are in the farm after $1,5,$ and $10$ months?

$($c$)(3$ points$)$ Is this harvesting strategy of $2000$ salmon per month sustainable in the long term without driving the salmon population to extinction? Justify your answer.