We consider the problem of finding a common fixed point of an infinite family ${T_n}$ of nonlinear self-mappings of a closed convex subset $C$ of a real Hilbert space $H$. Namely, we want to find a point $x$ with the property (assuming such common fixed points exist): [ xin igcap_{n=1}^infty ext{Fix}(T_n). ] We will use the Krasnoselskii-Mann (KM) Type inertial iterative algorithms of the form $$ x_{n+1} = ((1-alpha_n)I+alpha_nT_n)y_n,quad y_n = x_n + eta_n(x_n-x_{n-1}).eqno(*)$$ We discuss the convergence properties of the sequence ${x_n}$ generated by this algorithm (*). In particular, we prove that ${x_n}$ converges weakly to a common fixed point of the family ${T_n}$ under certain conditions imposed on the sequences ${alpha_n}$ and ${eta_n}$.