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Future value of an annuity Using the values below, answer the questions that follow. $($Click on the icon here in order to copy the contents of the data table below into a spreadsheet.$)$

$\backslash$table$[[$Amount of annuity,Interest rate,Deposit period $($years$)],[$$$6,000,9\%,10]]$

a$.$ Calculate the future value of the annuity, assuming that it is

$(1)$ An ordinary annuity.

$(2)$ An annuity due.

b$.$ Compare your findings in parts $a(1)$ and $a(2).$ All else being identical, which type of annuity$-$ordinary or annuity due$-$is preferable as an investment? Explain why.

a$.(1)$ The future value of the ordinary annuity is $

$($Rourfy to the nearest cent.$)$

$(2)$ The future value of the annuity due is $

$($Round to the nearest cent.$)$

b$.$ Compare your findings in parts $a(1)$ and $a(2).$ All else being identical, which type of annuity is preferable as an investment? $($Select the best answer below.$)$

Ordinary annuity, because it yields a greater future value.

Annuity due, because it yields a greater future value.

Gabrielle just won $$2$ million in the state lottery. She is given the option of receiving a of $$800,000$ now, or she can elect to receive $$100,000$ at the end of each of the next $20$ years. If Gabrielle can earn $9\%$ annually on her investments, which option should she take?

If Gabrielle takes the prize as an annuity, the present value of the $20-$year ordinary annuity is $ $($Round to the nearest dollar.$)$ If Gabrielle takes the prize as a single amount, the present value of the lump sum is $ $($Round to the nearest dollar.$)$ Which alternative should be chosen? $($Select the best answer below.$)$

A$.$ Lump sum payment

B$.$ Annuity payments