Mr$.$ Smith has received a large inheritance.$$ He wants to invest enough of the inheritance today to live off of for the next twenty years.$$ He is planning on the following expenditures for the next twenty years:

at the end of each of the first five years $($t $=1,2,3,4,5),$ $$50,000$ per year

at the end of each of the next ten years $($t $=6,7,8,9,10,11,12,13,14,15),$ $$75,000$ per year

at the end of each of the next three years $($t $=16,17,18)$ $$25,000$ per year

at the end of each of years $19$ and $20,$ $$85,000$ per year.

Also at the end of year $20,$ Mr$.$ Smith wishes to make a one$-$time payment of $$100,000$ to his nephew for graduate school.

A$.$ Based on these planned expenditures, how much must Mr$.$ Smith invest today to be able to cover these costs.$$ Assume that he can invest at a rate of $5\%$ per year.

B$.$ What effect would an increase in interest rates have on his required investment amount?$$ Why?

C$.$ What effect would a decrease in interest rates have on his required investment amount?$$ Why?