# Re-arranging equation: negative time for exponential

1. Homework Statement
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.

## Answers and Replies

mfb
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.

Analytical, how so?

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

mfb
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.

Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
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.

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.

mfb
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).