Solving Interest Problems: Compound Interest & PERT

[Pinu7]

Homework Statement

This is not a problem in my calculus book.However, I am sure this involves calculus. This is also not a question from an economics class, it is just curiosity.

My question is: If I have a debt that is continually compounded, and I continually pay off the debt at a constant rate, how long will it take to pay off the debt?

Homework Equations

Compund interest(PERT)

The Attempt at a Solution

Let:
r=rate on the debt. (assume annually)
y= amount of money I will pay per year.
\Deltat= an increment of time of which I will pay a quanta of money.

During the time \Delta t since I started the debt, I will owe e^r\Deltat

At this point I will pay my first quanta of money which would be y\Deltat. and w

Right before I make my second payment on time 2\Deltat, I will owe the money f
money owed from last increment AND the compound interest since that time.
ie I will owe (e^t\Deltat-y\Deltat)e^r\Deltat=e^2r\Deltat-y\Deltate^\Deltat

Continuing the pattern, the money I would owe right before my nth payment is:

e^n\Deltat-y\Deltate^(n-1)\DeltatThis is getting a bit tough. Where do I go from here?

 

[rootX]

Latex isn't working right now. It's bit hard to read.

But are you referring to a cash flow problem (annuities) i.e. there is a negative cash flow at t =0, and at the end of each year/month, there are equal positive cash flows.
http://www.zenwealth.com/BusinessFinanceOnline/TVM/Annuities.html(it doesn't consider inflation)