Calculate the following: $($ LG $15-2)$

a$.$ Suppose a $65-$year$-$old person wants to purchase an annuity from an insurance company that would pay $$20,000$ per year until the end of that person's life. The insurance company expects this person to live for $15$ more years and would be willing to pay $6$ percent on the annuity. How much should the insurance company ask this person to pay for the annuity?

b$.$ A second $65-$year$-$old person wants the same $$20,000$ annuity, but this person is healthier and is expected to live for $20$ more years. If the same $6$ percent interest rate applies, how much should this healthier person be charged for the annuity?

c$.$ In each case, what is the new purchase price of the annuity if the distribution payments are made at the beginning of the year?