When making large purchases, you usually pay a certain amount of the purchase up front $($called a down payment$)$ and then take a loan to pay off the remaining amount. The debt associated with said loan accrues interest which you also have to pay off, which is $($one of many ways$)$ how banks make money.

Say you bought a car for $$25,000$ from a dealership using a $24$ month payment plan and that the bank financing your loan charges interest at an annul rate of $5\%$ and compounds interest daily.

$($a$)$ Say you made no down payment. How much would you owe the bank by the end of the $24$ month payment period? Divide this amount by $24$ to find how much you would owe the bank each month. Hint: If you made no down payment, then all $$25,000$ dollars you borrowed is accruing interest.

$($b$)$ Write a function $f(n)$ that tells you how much you've payed for the car after $n$ months assuming you made no down payment.

$($c$)$ Say you made a $20\%$ down payment. How much would you owe the bank by the end of the $24$ month payment period? Divide this amount by $24$ to find how much you would owe the bank each month. Hint: If you made a $20\%$ down payment, then you would be borrowing $80\%$ of $$25,000$ dollars.

$($d$)$ Write a function $g(n)$ that tells you how much you've payed for the car after $n$ months assuming you made a $20\%$ down payment.