  The main objective of this paper is to connect the theory of $Gr\"obner$
bases to concepts of homological algebra. $Gr\"obner$ bases, an important tool
in algebraic system and in linear algebra help us to understand the structure
of an algebra presented by its generators and relations by constructing a basis
of its set of relations. In this paper we mainly deal with graded augmented
algebras. Given a graded augmented algebra with its generators and relations,
it is possible to construct a free resolution from its $Gr\"obner$ basis, known
as Anick's resolution. Though rarely minimal, this resolution helps us to
understand combinatorial properties of the algebra. The notion of a $K_2$
algebra was recently introduced by Cassidy and Shelton as a generalization of
the notion of a Koszul algebra. We compute the Anick's resolution of $K_2$
algebra which shows several nice combinatorial properties. Later we compute
some derived functors from Anick's resolution and outline how to construct
$A_\infty$-algebra structure on $Ext$-algebra from a minimal graded projective
resolution in which we can use Anick's resolution as a tool. Thus this paper
provides an unique platform to connect concepts of linear algebra with
homological algebra and homotopical algebra with the help of $Gr\"obner$ bases.
