A student borrows $\$ 3000$ on a bank credit card at a nominal rate of $18$% per year, which is actually charged at a rate of $1.5$% per month.
a) If a person does not place any additional charges on the card and pays the bank $\$150$ each month to pay off the loan. Let $B_n$ be the balance owed on the card after $n$ months. Find a closed formula for $B_n$.
b) How long will be required to pay off the debt?
HINTS
If his balance at the end of the $n$th month is $B_n$, can you find $B_{n+1}$ in terms of $B_n$? Remember to include another month's interest and his monthly payment. Now solve this recurrence relation with $B_0 = 3000$ to get a closed formula for $B_n$.
Solving $B_N = 0$ for $N$ will let you know when the payoff will be complete.