Question

Solve for $n$:
$\frac{1}{(n-1)(n-2)}+\frac{1}{(n-2)(n-3)}=\frac{2}{3},n\ne 1,2,3,$

Answer 1

$\to \frac{1}{n-2}(\frac{1}{n-1}+\frac{1}{n-3})=\frac{2}{3}$

$\Rightarrow \frac{n-1+n-3}{(n-2)(n-1)(n-3)}=\frac{2}{3}$

$\Rightarrow (2n-4)3=2(n-1)(n-2)(n-3)$

$\Rightarrow (n-2)(6-2(n-1\left)\right(n-3\left)\right)=0$

AS $n\ne 2,\Rightarrow 2(n-1)(n-3)=6$

$\Rightarrow n^2-4n+3=3$

$\Rightarrow n^2-4n=0$

$\Rightarrow n(n-4)=0$

$\Rightarrow n=0$ or $n=4$