For each $n \in \mathbb {N}$ prove that we can partition set $\{1,2,...,2^n\}$ into subsets $A$ and $B$ so that for each $0\le k \le n-1$ we have: $$ \sum_{x \in A} x^k = \sum_{x \in B} x^k $$
For this question, I thought of using mathematical induction and binomial expansion but I don't know how to use it here.
I would really appreciate it if someone could help me out.