Let R be a local commutative n-Gorenstein ring. The existence of the Gorenstein projective preenvelopes for finite R-modules is known (it was proved using duality arguments). In the present article, we compute an explicit Gorenstein projective preenvelope and a right Gorenstein projective resolution of a finite R-module. In light of this knowledge, we consider left derived functors Exti(−, −), Exti(−, −), and Gexti(−, −). We prove a balance result for the Tate derived functor Exti(−, −). Finally, we get an exact sequence connecting these derived functors.