This question is from Rotman's Introduction to the Theory of Groups:
(i) Suppose we have an exact sequence of free abelian groups $A\to B\to C\to D$ with maps $f,g,h$ in between. Show $B\cong \text{im}( f)\oplus \ker (h)$.
(ii) Given an exact sequence of free abelian groups ($n\geqslant 1$)
$$0\to F_n\to \cdots\to F_1\to F_0\to0$$
Show $\sum_{i=0}^n(-1)^i\text{rank}(F_i)=0$.
It's probably worth mentioning that on (ii), the book actually says $\sum_{i=0}^n\text{rank}(F_i)=0$, which I'm assuming is a typo because it fails for the sequence $0\to\mathbb{Z}\to\mathbb{Z}\to0$ with the identity in the middle. I'm just taking a guess on my correction because that's clearly true for $n=1$ and $n=2$.
Anyways, (i) is simple, just note that $B/\ker (g)\cong\text{im}(g)$ which is a subgroup of a free abelian group and hence free. Therefore $$B\cong \ker(g)\oplus(B/\ker(g))\cong\ker(g)\oplus\text{im}(g)=\text{im(}f)\oplus\ker(h)$$
I'm struggling with (ii). I don't really know where to begin. I wanted to try an induction argument but I don't see a way I can get a shorter exact sequence of the same form from this one. I think I should probably be using (i), but I'm not sure where I can apply it. Any ideas?
For (ii), try splitting your exact sequence into two exact sequences of the form $$0\to K \to F_{n-2}\to \cdots\to F_1\to F_0\to0$$ and $$0\to F_n\to F_{n-1}\to K \to 0.$$