Let $X$ be an algebraic variety and $F_1,...,F_n$ be a collection of coherent sheaves on $X$. Suppose we have a long exact sequence
$$0\to F_n\to F_{n-1}\to...\to F_1\to0.$$
Knowing all sheaf cohomologies $H^i(X, F_{j})$, $j=2,...,n$, how could I compute $H^i(X, F_{1})$?