# Solving for exponent and denominator

Is it possible to solve for $$r$$ in this equation?

$$v = \frac{1 - r^n}{1 - r}$$

I think it's not... I spent some time trying to wrestle with various substitutions and couldn't get there. I can solve for $$r$$ practically by writing a program to interpolate but I'm wondering if I'm missing something.

If it's not solvable, is there something about the form of the equation I should be able to recognize to know it's not solvable?

(Trivia: This is actually for a choose-your-own-adventure or "branching" novel; wondering whether I can solve for "choices per chapter" r given a number of chapters v and an average thread length n.)

• You are tacitly asking whether $\sum_{m=1}^{n-1}r^m=v$ has a solution. It has $n-1$ solutions. If you are asking whether one can find those solutions in closed form, the answer is yes for $n-1\le 4$. I doubt that $r^n-vr+v-1=0$ has a closed form solution. Oct 31, 2022 at 19:49
• This is equivalent to solving a polynomial equation, as we have $$1-r^n=(1-r)(1+r+r^2+\cdots+r^{n-1})\\ \implies v=\frac{1-r^n}{1-r}=1+r+r^2+\cdots+r^{n-1}$$ Oct 31, 2022 at 19:51
• @AndrewChin That expression is a tautology. It is true for all $r$. Oct 31, 2022 at 19:52
• Nov 1, 2022 at 0:15
• Yes, math.stackexchange.com/questions/902720/… answers my question as well. And teaches me this is a geometric sum. Thank you. Approximation is the way to go. Nov 2, 2022 at 17:53