Euler characteristic, what is the precise relationship between $\chi(V)$, $\chi(V')$, and $\chi(V'')$?

Question

For a graded vector space $V = \{V_n\}$ with $V_n = 0$ for all but finitely many $n$ and with all $V_n$ finite dimensional, define the Euler characteristic $\chi(V)$ to be $\sum (-1)^n \dim V_n$. Let $V'$, $V$, and $V''$ be such graded vector spaces and suppose there is a long exact sequence$$\dots \to V_n' \to V_n \to V_n'' \to V_{n-1}' \to \dots.$$I am curious, what is the precise relationship between $\chi(V)$, $\chi(V')$, and $\chi(V'')$?