The company wishes to sell its car for $23000$ usd cash. A client who wants the car offers to make semi-annual payments with a rate of $11.5\%$ with semi-annual capitalization, the first at the end of the first semester and the last payment at the end of four and a half years, and one year later pay the sum of $5000$ usd. If the company accepts these conditions, calculate the value of the semester fee.
I have solved but I am not sure if the answer is correct or not.
My solution:
Principal $P=23000$
rate $=\frac{11.5}{100}=0.015$
Semi-annual rate $r=\frac{0.015}{2}=0.0575$
last year's payment $f=5000$
Considering that four and a half years is equivalent to $10$ semesters then the total of semsters $n=10$
Then,
$$P=f+\text{semester fee}\,\sum_{k=1}^n(1+r)^{-k}.$$
So we have to
$$\text{semester fee}= \frac{P-f}{\sum_{k=1}^n(1+r)^{-k}}=\frac{23000-5000}{\sum_{k=1}^{10}(1+0.0575)^{-k}}=1038.88.$$