Problem

Source: Albanian Mathematical Olympiad 12 GRADE 2011--Question 4

Tags: induction, algebra unsolved, algebra



The sequence $(a_{n})$ is defined by $a_1=1$ and $a_n=n(a_1+a_2+\cdots+a_{n-1})$ , $\forall n>1$. (a) Prove that for every even $n$, $a_{n}$ is divisible by $n!$. (b) Find all odd numbers $n$ for the which $a_{n}$ is divisible by $n!$.