Exponencial grow, with controlable outcome and steps.

Question

I'm not a mathematician so I'm not sure how to calculate what I need. The problem is, I want to go from 0 to 0.9, growing exponentially, over a determined amount of steps. And with every step added together, the overall result must be 9. Is there a way to calculate this? thanks

Answer 1

Your description of your problem is not clear. If $\ q\ $ is the quantity you want to grow exponentially, do you want it to grow continuously or in discrete steps. Your phrase "over a determined amount of steps" would seem to suggest the latter, but it's certainly not obvious that that's what you mean.

A quantity which grows exponentially in discrete steps will have the form $\ s(1+g)^n\ $ after $\ n\ $ steps, where $\ s\ $ is the initial value of the quantity, and $\ g\ $ is the fraction by which it grows at each step. If you add all those values together, you'll get $$ s\sum_{i=1}^n(1+g)^i=s\left(\frac{(1+g)^n-1}{g}\right)\ . $$ If you start with $\ s=0\ $ this quantity will remain zero no matter how many steps you take, so you could never reach any positive value. If you start with a positive value for $\ s\ $, however, and $\ N\ $ is the "determined" number of steps you have available, then you will need $$ 0.9=s\left(\frac{(1+g)^N-1}{g}\right)\ , $$ or, equivalentlty, $$ (1+g)^N-\left(\frac{0.9}{s}\right)g-1=0\ . $$ For any fixed $\ N\ $ and $\ s>0\ $, this equation will have a unique positive solution for $\ g\ $, and this growth rate will achieve your aim of "going from" $\ s\ $ "to $\ 0.9\ $" in $\ N\ $ steps, provided $\ s>0\ $. In general, there will be no simple formula for the solution, however. While it will be possible to obtain arbitrarily close numerical approximations to the solution, that's the best you can hope for in the general case.