Solve the following equation:

$$
\frac{x}{a^{n-1} b^{n}}-\frac{x}{a^{n} b^{n-1}}=\frac{1}{b^{n}}-\frac{1}{a^{n}}
$$
Multiply both sides of the equation by $a^{n} b^{n}$. We get

$$
a x-b x=a^{n}-b^{n},
$$

that is,

$$
x=\frac{a^{n}-b^{n}}{a-b}=a^{n-1}+a^{n-2} b+\ldots+a b^{n-2}+b^{n-1}
$$

(Jenö Silbermann, Nagyvárad.)