# Interest problem

## 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 er$$\Delta$$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 (et$$\Delta$$t-y$$\Delta$$t)er$$\Delta$$t=e2r$$\Delta$$t-y$$\Delta$$te$$\Delta$$t

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

en$$\Delta$$t-y$$\Delta$$te(n-1)$$\Delta$$t

This is getting a bit tough. Where do I go from here?

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