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.
$$\Delta$$t= 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$$\Delta$$t

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$$\Delta$$t, I will owe the money f
money owed from last increment AND the compound interest since that time.
ie I will owe (e^t$$\Delta$$t-y$$\Delta$$t)e^r$$\Delta$$t=e^{2r$$\Delta$$t}-y$$\Delta$$te^$$\Delta$$t

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

e^n$$\Delta$$t-y$$\Delta$$te^{(n-1)$$\Delta$$t}This 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)