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Quillen's Resolution Theorem in algebraic $K$-theory provides a powerfulcomputational tool for calculating $K$-groups of exact categories. At the levelof $K_0$, this result goes back to Grothendieck. In this article, we firstestablish an extriangulated version of Grothendieck's Resolution Theorem.Second, we use this Extriangulated Resolution Theorem to gain new insight intothe index theory of triangulated categories. Indeed, we propose an index withrespect to an extension-closed subcategory $\mathscr{N}$ of a triangulatedcategory $\mathscr{C}$ and we prove an additivity formula with error term. Ourindex recovers the index with respect to a contravariantly finite, rigidsubcategory $\mathscr{X}$ defined by J{\o}rgensen and the second author, aswell as an isomorphism between $K_0^{\mathsf{sp}}(\mathscr{X})$ and theGrothendieck group of a relative extriangulated structure on $\mathscr{C}$ when$\mathscr{X}$ is cluster tilting. In addition, we generalize and enhance someresults of Fedele. Our perspective allows us to remove certain restrictions andsimplify some arguments.

A resolution theorem for extriangulated categories with applications to the index