KöMaL 2017 B4883

Define the sequence $a_1, a_2, \dots$ with the following recurrence relation: $a_1=4, a_2=2 \text{and} a_{n+1}=\frac{na_n^2}{na_n^2-(n+1)a_n+n+1}, \text{ if } n\ge 2.$ Prove that $a_1+2\cdot a_2+3\cdot a_3+\cdots +n\cdot a_n=a_1\cdot a_2\cdot a_3\cdot\,\cdots \cdot a_n$ for all $n\ge 1$.