A loan of $17,000$ dollars is to be repaid in annual installments of $2,100$ dollars, the first due in one year, followed by a final smaller payment. If the effective rate of interest is $8.8$ percent, what is the outstanding balance owed immediately after the $5th$ payment?
Im pretty confused on how to go about solving this, I know that outstanding balance is equal to the payment amount times the present value at a specified time, but I am confused how to go about setting up the equation properly.
The future value of the loan after 5 years is $17,000\cdot(1.088)^5$.
The future value of the 5 payments is
$C_5=2100\cdot (1.088)^4+2100\cdot (1.088)^3+2100\cdot (1.088)^2+2100\cdot (1.088)^1+2100\cdot (1.088)^0$
The first payment has to be compounded 4 times.
The last payment has not be compounded.
$2100\cdot \left[ (1.088)^4+ (1.088)^3+ (1.088)^2+2100\cdot (1.088)^1+ (1.088)^0 \right]$
The expression in the brackets is the partial sum of a geometric series. Therefore
$C_5=17,000\cdot(1.088)^5-2100\cdot \frac{1-1.088^5}{1-1.088}$
I hope it helps.