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## 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.

[tex]\Delta[/tex]t= an increment of time of which I will pay a quanta of money.

During the time [tex]\Delta[/tex] t since I started the debt, I will owe e

^{r[tex]\Delta[/tex]t}

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

Right before I make my second payment on time 2[tex]\Delta[/tex]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[tex]\Delta[/tex]t}-y[tex]\Delta[/tex]t)e

^{r[tex]\Delta[/tex]t}=e

^{2r[tex]\Delta[/tex]t}-y[tex]\Delta[/tex]te

^{[tex]\Delta[/tex]t}

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

e

^{n[tex]\Delta[/tex]t}-y[tex]\Delta[/tex]te

^{(n-1)[tex]\Delta[/tex]t}

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

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