According to the United Press International, honeybee keepers lose 30% of their colonies each and every winter due to unknown illness. They also lose 5% of what remains because of other issues. Carlisle Honey reports that each spring, 4690 bees are born (no matter what happened over the winter). Assume at the start of summer 2010, Carlisle Honey had 6000 bees.

a. Write down a recursive equation describing the number of bees owned by Carlisle Honey at the start of year n (where n = 0 corresponds to 2010).

b. Find the number of bees at the start of summer 2020.

c. Over time, the number of bees will tend to approach what value?

d. Now choose a different number of bees to start the problem (your choice, but tell us what it is), and suppose the company still loses 30% each winter due to an unknown illness, loses 5% because of other reasons, and gets 4690 new bees each spring. Over time, the number of bees will approach what value?

e. Based upon your answers to (c) and (d), make a conjecture about the effect of the initial number of bees on the number of bees over time. Prove your conjecture.