Let a be an ideal of a commutative noetherian ring R. In 1970, Hartshorne posed two Questions on the finiteness of local cohomology modules: Are the local cohomology modules H}^{i\geq 0}_a(M) {a}-cofinite for every finitely generated R-module M? Is it true that an R-complex X is a-cofinite iff its homology module H_i(X) is a-cofinite for every i? In this paper, we prove some new results concerning theses Questions by using a spectral sequence argument.