From Rotman's Algeraic Topology:
If $0 \rightarrow A_n \rightarrow A_{n-1} \rightarrow \dots \rightarrow A_1 \rightarrow A_0 \rightarrow 0$ is an exact sequence of (finitely generated) abelian groups, then $\sum_{i=0}^n(-1)^i \text{ rank }A_i = 0$.
I'm having trouble figuring out how to start this. I've looked at induction on the number of Abelian groups (proven true for $n=0,1,2$), using the first isomorphism theorem, and extending the maximally independent subsets but I keep getting stuck.
Anyone have any ideas?