Take an exact sequence of $A$-modules $$0 \rightarrow M_r \rightarrow M_{r-1} \rightarrow \dots \rightarrow M_0 \rightarrow 0$$
Prove that $\sum_{0\le i\le r} (-1)^i$ len$_A M_i = 0$.
I know how to prove the result if it's a short exact sequence, but i have no idea how to progress from here. Maybe some sort of induction?
Hint:
Factor $M_2\longrightarrow M_1$ through its image $E_1\subset M_1$, so that you obtain the exact sequence $$0 \longrightarrow M_r \longrightarrow M_{r-1} \longrightarrow \dots \longrightarrow M_2\longrightarrow E_1\longrightarrow 0$$ and the short exact sequence: $\quad0\longrightarrow E_1\longrightarrow M_1\longrightarrow M_0\longrightarrow 0$.
