Reduced homology groups of a space which is the union of finitely many open subsets whose intersection is empty or has trivial reduced homology.

Question

This is exercise 33 (p.158) from section 2.2 in Hatcher's Algebraic Topology:

Suppose the space $X$ is the union of open sets $A_1, \ldots, A_n$ such that each intersection $A_{i_1} \cap \cdots \cap A_{i_k}$ is either empty or has trivial reduced homology groups. Show that $\tilde{H_{i}}(X) = 0$ for $i \geq n-1$.

I'm pretty sure Mayer-Vietoris needs to be used, and that there should be some induction going on, but I haven't been able to figure it out.