  Quillen's Resolution Theorem in algebraic $K$-theory provides a powerful
computational tool for calculating $K$-groups of exact categories. At the level
of $K_0$, this result goes back to Grothendieck. In this article, we first
establish an extriangulated version of Grothendieck's Resolution Theorem.
  Second, we use this Extriangulated Resolution Theorem to gain new insight
into the index theory of triangulated categories. Indeed, we propose an index
with respect to an extension-closed subcategory $\mathscr{N}$ of a triangulated
category $\mathscr{C}$ and we prove an additivity formula with error term. Our
index recovers the index with respect to a contravariantly finite, rigid
subcategory $\mathscr{X}$ defined by J{\o}rgensen and the second author, as
well as an isomorphism between $K_0^{\mathsf{sp}}(\mathscr{X})$ and the
Grothendieck group of a relative extriangulated structure
$\mathscr{C}_{R}^{\mathscr{X}}$ on $\mathscr{C}$ when $\mathscr{X}$ is
$n$-cluster tilting. In addition, we generalize and enhance some results of
Fedele. Our perspective allows us to remove certain restrictions and simplify
some arguments.
  Third, as another application of our Extriangulated Resolution Theorem, we
show that if $\mathscr{X}$ is $n$-cluster tilting in an abelian category, then
the index introduced by Reid gives an isomorphism
$K_0(\mathscr{C}_R^{\mathscr{X}}) \cong K_0^{\mathsf{sp}}(\mathscr{X})$.
