A certain protected species normally doubles its population each month. The initial

population $($at the beginning of month $1)$ is $20,$ but by the beginning of the next month, $1$

specimen has died of an infection. In successive months, the infection kills $2,$ then $4,$

then $8,$ and so forth. In table form, this is

Beginning of Month Number killed

$10$

$21$

$32$

$44$

$58$

a$.$ Write a recurrence relation for the population at the beginning of month n$.$

b$.$ Solve this recurrence relation using the result of Problem $1.$ The result should be

an explicit expression involving n$,$ not an order of magnitude expression.

c$.$ What is the size of the population at the beginning of month $12?$

d$.$ At the beginning of what month has the population died out?