Find compound growth rate from cumulative totals I'm a bit out of my depth here, so please feel free to correct any errors in terminology, etc.
I'm looking to solve for a percentage growth rate. I know the starting population, the number of compounding periods, and the cumulative total of the population.
When I say cumulative, I mean the sum total of each compounding period. An example:


*

*Starting population: 100

*Compounding periods: 3

*Cumulative total: 331
In this case, the growth rate turns out to be 10%, because:


*

*Period 1: 100

*Period 2: 110

*Period 3: 121

*Cumulative Total: 100 + 110 + 121 = 331
What I'm looking for is an equation that describes this relationship, like:
Percent growth = ...
Any ideas?

The cummulative total is $S_n=100+110+121$. It can be transformed by factoring out 1.1 and $1.1^2$.
$S_n=100+1.1\cdot 100+1.1^2\cdot 100$ This is the partial sum of a geometric series.
The formula for the partial sum of a geometric series is $S_n=C_0\cdot \frac{(1+g)^n-1}{g}$
$C_0$ is the starting population. $g$ is the growth rate. In your example $C_0=100,n=3$ and $S_n=100$
The formula can be transformed. $S_n\cdot g=C_0\cdot (1+g)^n-C_0$ This equation has to be solved for g.
If n=2, then you have a quadratic equation. It easy to solve.
In your case it is n=3: Here you can apply Cardano´s method or use an approximation method like the Newton–Raphson method. Both methods are time-consuming.
