Notation : $Z_{i+1}(G)=${$g\in G : [g,x]\in Z_i(G) \forall x\in G$} where $Z_0(G)=1$.
Definition : $G$ is said to be nilpotent if $Z_n(G)=G$ for some $n\in \Bbb N$.
Let $Z_n(G/Z(G))=G/Z(G)$.
I have that $Z_{i+1}(G)/Z_i(G)=Z(G/Z_i(G))$. Then $\frac{Z_{i+1}(G)/Z(G)} {Z_i(G)/Z(G)}≈Z(\frac{G/Z(G)}{Z_i(G)/Z(G)})$. Now the L.H.S is a subgroup of $\frac{G/Z(G)}{Z_i(G)/Z(G)}$ and the R.H.S is a characteristic subgroup of $\frac{G/Z(G)}{Z_i(G)/Z(G)}$. Since L.H.S is isomorphic to R.H.S, I conclude that L.H.S.=R.H.S. At this stage I am stuck. What can I conclude from this?