How can we solve the following equation for $x$ without a calculator/computer? ($m$ and $n$ are positive integers)
$m^{n}(1-\frac{x}{m})(1-\frac{x}{m-1})(1-\frac{x}{m-2})=1.$
I tried to estimate $1-\frac{x}{m}$ by $e^{-\frac{x}{m}}$ and do the same for the rest, but I got stuck.
Computer algebra gives the solution:
$$x = -\frac{\sqrt[3]{27 m^{2 n}+54 m^{2 n-2}-81 m^{2 n-1}+\sqrt{\left(27 m^{2 n}+54 m^{2 n-2}-81 m^{2 n-1}\right)^2-108 m^{6 n-6}}} m^{1-n}}{3 \sqrt[3]{2}}-\frac{\sqrt[3]{2} m^{n-1}}{\sqrt[3]{27 m^{2 n}+54 m^{2 n-2}-81 m^{2 n-1}+\sqrt{\left(27 m^{2 n}+54 m^{2 n-2}-81 m^{2 n-1}\right)^2-108 m^{6 n-6}}}}+m-1$$
which suggests that you'll not easily find this solution by pen and paper.