KöMaL 2005 B3817

The sequence $a_n$ is defined as follows: $a_1 \ne$2, $a_2=\frac{a_1}{a_1-2}$, $a_{n+1}=\frac{na_n^2}{na_n^2-(n+1)a_n+(n+1)}$, if $n \ge$2. Prove that the inequality $a_1$+2$a_2$+3$a_3$+...+$na_n$=$a_1 ^. a_2 ^. a_3 ^. ...^. a_n$ is true for all $n$.