= Solution
Use the <commutator> convention $[x,y]=x^{-1}y^{-1}xy$, and let $[A,B]$ denote the <subgroup> generated by all such commutators with $x\in A,y\in B$. The <lower central series> is
$$
\gamma_1(G)=G,\qquad\gamma_{i+1}(G)=[\gamma_i(G),G].
$$
A <central series> has $[G_j,G]\le G_{j-1}$ for each $j\ge1$. We prove the requested inclusion by induction. At $i=0$, $\gamma_1(G)=G_n$. If $i<n$ and $\gamma_{i+1}(G)\le G_{n-i}$, then
$$
\gamma_{i+2}(G)=[\gamma_{i+1}(G),G]
\le[G_{n-i},G]\le G_{n-i-1}.
$$
Therefore
$$
\boxed{\gamma_{i+1}(G)\le G_{n-i}\quad(0\le i\le n).}
$$
In particular $\gamma_{n+1}(G)=1$. The proof uses the central-series condition directly; no equality of the intermediate terms is claimed.
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