$($Comprehensive problem$)$ You would like to have $$51,000$ in $16$ years. To accumulate this amount, you plan to deposit an equal sum in the bank each year that will earn $8$ percent interest compounded annually. Your first payment will be made at the end of the year.

a$.$ How much must you deposit annually to accumulate this amount?

b$.$ If you decide to make a large lump$-$sum deposit today instead of the annual deposits, how large should this lump$-$sum deposit be$?($Assume you can earn $8$ percent on this deposit.$)$

c$.$ At the end of five years, you will receive $$10,000$ and deposit this in the bank toward your goal of $$51,000$ at the end of year $16.$ In addition to the lump$-$sum deposit, how much must you deposit in equal annual amounts, beginning in year $1$ to reach your goal? $($Again$,$ assume you can earn $8$ percent on your deposits.$)$

a$.$ How much must you deposit annually to accumulate this an ${?}^{\text{'}}$ no$\text{'}...1$ t$?$

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

b$.$ If you decide to make a large lump$-$sum deposit today instead of the annual deposits, how large should the lump$-$sum deposit be$?$

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

c$.$ If you deposit $$10,000$ received at the end of five years in the bank, what will the amount grow to by the end of year $16?$

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

In addition to the lump$-$sum deposit, how much must you deposit in equal annual amounts, beginning in Year $1$ to reach your goal? pleaes answer all questions, thank you!