A3. The expression is $\frac{x^{n-1}}{x^{n}-2 x^{n-1}}-\frac{x^{n}}{x^{n+1}-4 x^{n-1}}$. Which expression is equivalent to the given expression for $x \neq 0$?
(A) $\frac{1}{(x-2)}$
(B) $\frac{2}{(x-2)(x+2)}$
(C) $\frac{1}{(x+2)}$
(D) $\frac{2 x}{(x-2)(x+2)}$
(E) $\frac{1-x}{(x-2)(x+2)}$

Problems for 1st year
A3. In the denominators of the fractions, we factor out the common factor and simplify where possible $\frac{x^{n-1}}{x^{n}-2 x^{n-1}}-$ $\frac{x^{n}}{x^{n+1}-4 x^{n-1}}=\frac{x^{n-1}}{x^{n-1}(x-2)}-\frac{x^{n}}{x^{n-1}\left(x^{2}-4\right)}=\frac{1}{(x-2)}-\frac{1}{x^{-1}\left(x^{2}-4\right)}$. We add the fractions $\frac{1}{(x-2)}-\frac{x}{\left(x^{2}-4\right)}=$ $\frac{x+2}{(x-2)(x+2)}-\frac{x}{(x-2)(x+2)}=\frac{2}{(x-2)(x+2)}$. The correct answer is $B$.