Question 198
Given a positive integer $m$, a sequence of real numbers $a = (a_1, a_2, a_3, ...)$ is called $m$-powerful if it satisfies
$$
\left( \sum_{k=1}^{n} a_k \right)^m = \sum_{k=1}^{n} a_k^m \quad \text{for all positive integers } n.
$$
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 is zero but which is 2017-powerful.
$$
\left( \sum_{k=1}^{n} a_k \right)^m = \sum_{k=1}^{n} a_k^m \quad \text{for all positive integers } n.
$$
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 is zero but which is 2017-powerful.