# Re-arranging equation: negative time for exponential

1. Apr 20, 2013

### jackscholar

1. The problem statement, all variables and given/known data
I have a sequence whereby
10000=100(1+e^(kt)+e^(2kt)+...+e^(39kt)) where k=-4.7947012×10^(-3) which was dervied from dy/dt=ky
Re-arranging i get 99=1+e^(kt)+e^(2kt)+...+e^(39kt), letting e^(kt)=r I put it into the computer and
i get 1.04216=r=e^-4.7947012×10^(-3)t
taking ln of both sides and dividing by the number gives a negative time value. Any help is highly appreciate.

2. Apr 20, 2013

### Staff: Mentor

With a negative k, t has to be negative, too. That is a correct solution of the equation. If a negative time value is impossible, your equation (or k) has to be wrong.

There is an analytic solution, by the way.

3. Apr 20, 2013

### jackscholar

Analytical, how so?

4. Apr 20, 2013

### jackscholar

Is it by stating that, if assuming that all doses haven't decayed, the maxmimum amount is 3900 only.

5. Apr 20, 2013

### Staff: Mentor

With r=e^(kt), your equation gets 99=r^0 + r^1 + r^2 + ... + r^39
That is a geometric expression, and has a nice formula.

6. Apr 20, 2013

### Ray Vickson

Right. And that gives you a degree = 40 polynomial to solve---not easy at all, and I doubt there is an analytical formula for its solution.

7. Apr 20, 2013

### Staff: Mentor

Hmm, you are right. Well, at least it is easier to solve it numerically that way (which does not matter if a computer solves it).