Present Value of loan payment You value a car to be $30,000. If you plan to make continuous payments over 5 years
and at an interest rate of r = 10%.
1) How much should you pay per year so that the present value of your total payments
in 30; 000?
The formula that i think i should use is
$$PV = PMT\frac{1-(1+i)^{-n}}{i}$$
So solving for $PMT$ I got 7913.48. Did i do that correctly?
2)What if instead you decided to let your payments increase with time and pay at a
rate of $6000 + t*1000$ per year, where t is measured in years. How long would it
take you to pay off the car ? (Note the equation you get might be difficult to solve,
so you can use a graphing calculator to estimate.)
I'm not too sure how to approach this. Do i use the same equation and solve for t?
I assume the $t$ is the same as the $n$ I used in my $pv$ formula, is this correct?
Thanks
 A: I do not know what $r$ reprents, so will assume that it is the effective annual rate. 
If that is the case, then a dollar grows in a year to $e^k$ dollars, where $e^k=1.10$. Thus $k=\ln(1.10)$. But we will keep on using $k$.
Let $c$ be the amount paid per year. So we are paying at a yearly rate of $c$. The present value of these payments is 
$$\int_0^5 c e^{-kt}\,dt.$$
Integrate. The result should be $30000$. Thus we obtain the equation
$$c\frac{1}{k}\left(1-e^{-5k}\right)=30000.$$
In the above equation, we know everything except $c$, so can solve for $c$.
For the second problem, the setup is roughly similar, except that we do not know the number $y$ of years, but we do know the rate of payment. We end up with the equation
$$\int_0^y (6000+1000t)e^{-kt}\,dt=30000.$$
We can do the integration using integration by parts. However, we end up with an equation in $y$ that we cannot solve analytically. However, the equation can be solved by using a numerical method. It can also be solved to adequate accuracy with a graphing calculator.
Remark: Conceivably the $r$ could refer to the force of interest. In that case we would have $k=0.10$. I chose to interpret it as the effective annual rate, since that seemed more plausible. 
