# Solve an equation with the form $y=\frac{x^n-1}{x-1} + c$ (finding $r$ in a geometric series)

The sum $S_n$ of the first $n$ terms of a geometric series is given by $S_n=\frac{a(r^n-1)}{r-1}$, where $a$ is the first term and $r$ is the common ratio.
How do you solve an equation with this following form (solve for $x$):
$$y=\frac{x^n-1}{x-1} + c$$ $n$ and $c$ are both real numbers. The above equation is another form of the geometric series formula, since $a=1$, $r=x$, and $S_n=-c$. Since it is similar to the sum of a geometric series formula, my question can also be interpreted as, how to find $r$ given $a$, $n$, and $S_n$ of a geometric series.
Methods like long division might not work since the values of $n$ can be very large. Is there an inverse form of the above equation?
Example question: Solve for $x$:
$$0=\frac{x^{36}-1}{x-1}-10$$

$10=\frac{x^{36}-1}{x-1}\qquad |\cdot (x-1)$
$10x-10=x^{36}-1$
$x^{36}-10x+9=0$
This is a polynomial equation of degree 36. This equation can not solved algebraically. $x=1$ could be a solution, but the first equation shows that $x\neq 1$ (denominator). You have to apply an approximation method, like the Newton-Raphson method, to solve the equation.