# Interest continuously compounding with a variable prinicple

1. Sep 9, 2011

### venik

1. The problem statement, all variables and given/known data
This is a question I *might* have already got the answer to, I'd just like for someone very good with calculus and algebra to verify/answer the question themselves.

To be more exact with the problem, we are putting 1 dollar (a variable principle) into a continuously compounded interest account at rate r. At what moment does one year's interest (e^r-1), equal our principle of 1 dollar per year. Or at what time can we stop putting the dollar in completely.

I solved it as the most complicated substitution problem I, personally, have ever done. Feel free to do it any way you please, but I'm looking for both answers and/or possible mistakes in my math.

2. Relevant equations
F=Pe^rt

3. The attempt at a solution

Let F = Final, P = principle, r = interest rate, y = years.
Given that:
F = Pe^(ry)

And

P1 = ($)1 x y (P1 because I'm going to have to use another P later) Then replacing 1y for P we get F = ye^(ry) This gives us F for any time y, and rate r. But we want a specific F, to get that I first defined what P2 is required for the next year's interest to be equal to the 1 dollar we are putting in every year. P2(e^r-1) = 1 P2=1/(e^r-1) In order to substitute this into our original equation we must substitute P2 into a separate F = Pe^(rt2) We know that t2 = 1 because in the question we asked when does (e^r-1) of the *last* year equal$1.

We get

F = e^r/(e^r-1)

Then substitute this final into our F = ye^ry

we get

e^r/(e^r-1) = ye^ry

0 = ye^ry - e^r/(e^r-1)

at 15% interest I get y = 3.96 years. Graphing on my calculator because as far as I know that is unsolvable algebraically

at 8% interest I get y = 7.27 years.

Sounds too good to be true. Put $20k in a savings account for 4-7 years and you will raise your wage$20k/year for the rest of your life? Start investing! It will be free soon. lol.

This is only my most recent approach to this problem. Other answers which I have debunked are (for 8% interest) 13.8 years, 20 years, and 7.5 years. I'd like confirmation, or a correct answer please.

2. Sep 12, 2011

### vela

Staff Emeritus
I think you haven't gotten any responses because it's not at all clear exactly what the set-up of the problem is.