Find parameters of short geometric series I occasionally host D&D~ish game and having a way to do this will definitely improve my ability to serve better games, so when they want to upgrade their fireball 5 times, I can do it without the game getting out of hand :)
There are four related numbers: Base $B = 100$, Start $S = 0.4$, Count $C = 4$, and Multiplier $M = 0.64448$, for example.  I need to be able to find $M$ by having $B$, $S$ and $C$. Also I need to find $S$ by having $B$, $C$ and $M$.
The way these four numbers are related is through a geometric series.  Let $r_0 = B\times S$, and let $r_{i+1} = M\times r_i$ for $0 \le i \lt C$ define $r_1,\ldots,r_C$.  It is desired that $B \approx r_0 + r_1 + \ldots + r_C$, where an error in approximation less than one is tolerable.
In the sample figures shown above:
(B) 100 × (S) 0.4 = 40 (r0)  
(r0) 40 × (M) 0.64448 = 25.7788368 (r1) .... (counting 1)  
(r1) 25.7788368 × (M) 0.64448 = 16.61371067 (r2) .... (counting 2)  
(r2) 16.61371067 × (M) 0.64448 = 10.7070534 (r3) .... (counting 3)  
(r3) 10.7070534 × (M) 0.64448 = 6.900384555 (r4) .... (counting 4)  

Adding $r_0 + r_1 + r_2 + r_3 + r_4$ should give $B$ or close to it (with the above numbers the sum is about $100.001$).
Keep in mind $B$, $S$ and $C$ aren't always the same.  Last time I needed it, $B = 1319143$, $S = 0.156048362$ , $M=0.85$, $C=19$ and the sum of $r_0 + r_1 + \ldots + r_{19}$ was about $1319142.86296796$ (within our margin of $\pm 1$).
Please use algebra, since my math knowledge if quite limited.
 A: let $b$ and $s$ denote the base and start variable you mention, and let $n$ be the count, and let $m$ be the multiplier.
If I understant correctly, you then commit $n$ steps to get $n$ results:
$$r_1 = b\cdot s \cdot m\\
r_2 = r_1\cdot m\\
r_3 = r_2\cdot m\\
r_4 = r_3\cdot m$$
Then you sum the results to get $100$.
Now, from my equations, it is simple to see that $r_2=b\cdot s \cdot m^2$, $r_3=b\cdot s \cdot m^3$ and $r_4=b\cdot s \cdot m^4$, so $$r_1+r_2+r_3+r_4 = b\cdot c\cdot (m + m^2 + m^3 + m^4),$$
and you want to solve the equation $r_1+r_2+r_3+r_4 = 100$ for $m$, so you are solving $$m+m^2+m^3+m^4 = \frac{100}{bc}.$$
In general, a polynomial equation of order $>2$ is hard to solve. In this case, the power sum $m+m^2+m^3+m^4$ can be simplified by using the fact that $$x+x^2+x^3+\dots + x^n = \frac{x(x^n-1)}{x-1}$$ holds for any integer $n$. This means that the equation you need to solve is $$\frac{m(m^4-1)}{m-1} = \frac{100}{bc}$$
Either way, the equation is solvable analytically for up to $4$ steps, but is (to my knowledge) not solvable analytically, but can easily be solved using numeric means.

On the other hand, you can very simply extract $s$ (the start) from this equation, as $$r_1+r_2+r_3+r_4 = 100$$ reduces to $$s\cdot b(m+m^2+m^3+m^4) = 100,$$
so $$s = \frac{100}{b(m+m^2+m^3+m^4)}$$
A: Algebra can help us see the real problem to be solved.
When we have a sequence of values to sum, there is a special symbol for it, the Greek letter Sigma (uppercase):
$$ \sum_{i=0}^C r_i = r_0 + r_1 + \ldots + r_C $$
In the special case described here we have terms in geometric progression.  That is, term $r_i$ is multiplier $M$ times the previous term $r_{i-1}$.
Using this notation the problem can be restated as:  Given $B,S,C$ find $M \gt 0$ such that
$$ B = B\cdot S + B\cdot S\cdot M + \ldots + B\cdot S\cdot M^C = \sum_{i=0}^C B\cdot S\cdot M^i = B\cdot S\cdot \sum_{i=0}^C M^i $$
We now see that the "base" parameter $B$ doesn't really interact with the other parameters.  We can divide both sides of the equation by $B$ and it disappears from the problem:
$$ 1 = S\cdot \sum_{i=0}^C M^i $$
If $C$ and $M$ are given, it's a simple matter to compute $S = (\sum_{i=0}^C M^i)^{-1}$.  If $M=1$, then $S = 1/C$.  Otherwise the formula for a finite geometric series gives:
$$ S = \frac{M-1}{M^{C+1}-1} $$
This could be evaluated with a few calculator keypresses.
The problem of finding $M$ given $S$ and $C$ is more difficult, because it requires finding the root $M$ of a polynomial of degree $C$:
$$ M^C + \ldots + M + 1 = S^{-1} $$
As a function of $M \gt 0$, the left-hand side of this polynomial equation is strictly increasing.  Therefore when $S^{-1} \gt 1$, we will have exactly one positive real root for $M$.  There is no simple solution "with a few calculator keypresses" but finding a numerical value (approximation) for $M$ is not difficult.
Knowing that $M=0$ is too small, we can easily find a value that is too large.  For example clearly $M=S^{-1}$ is too large.  The simplest root finding procedure is bisection: take an interval that brackets the unique positive root (upper bound and lower bound), and chop it in half.  Evaluate the polynomial above at the midpoint of that interval, and we find out whether the midpoint value is too small or too large.  We continue using the upper half or lower half of the previous interval accordingly.
There are numerical methods for solving this problem, but if calculus level math is not your strength, then this (systematic) "trial and error" approach is a pretty good approach.
With a bit of effort (and maybe help from one of your D&D'ish colleagues), an Excel spreadsheet or equivalent can be created to automate the trial-and-error.
