While studying Module Theory, I've been stuck on this exercise; any help would be very welcome:
Let $0 \longrightarrow M_1 \longrightarrow M_2 \longrightarrow ... \longrightarrow M_{n-1} \longrightarrow M_n \longrightarrow 0$ be an exact sequence of $R$-modules such that $lh(M_i)$, the lenght of $M_i$, is finite for all $i \in \{1,...,n\}$. Show that
$\sum^n_{i=1}(-1)^ilh(M_i)=0.$
I feel that one could argument by induction (on $n$), but I don't see how.