IrMO 2017 Problem 10

Given a positive integer $m$ , a sequence of real numbers $a= (a_1,a_2,a_3,...)$ is called $m$ -powerful if it satisfies $$(\sum_{k=1}^{n} a_k )^{m} = \sum_{k=1}^{n} a_k^{m}$$ for all positive integers $n$ . (a) Show that a sequence is $30$ -powerful if and only if at most one of its terms is non-zero. (b) Find a sequence none of whose terms are zero but which is $2017$ -powerful.