Large groups and their periodic quotients

We first give a short group theoretic proof of the following result of Lackenby. If $G$ is a large group, $H$ is a finite index subgroup of $G$ admitting an epimorphism onto a non-cyclic free group, and $g_1, \ldots, g_k$ are elements of $H$, then the quotient of $G$ by the normal subgroup generated by $g_1^n, \ldots , g_k^n$ is large for all but finitely many $n\in \mathbb{Z}$. In the second part of this note we use similar methods to show that for every infinite sequence of primes $(p_1, p_2, \ldots )$, there exists an infinite finitely generated periodic group $Q$ with descending normal series $Q=Q_0\rhd Q_1\rhd \ldots$, such that $\bigcap_i Q_i=\{ 1\}$ and $Q_{i-1}/Q_i$ is either trivial or abelian of exponent $p_i$.