  We study the problem of finding zero-sum blocks in bounded-sum sequences,
which was introduced by Caro, Hansberg, and Montejano. Caro et al. determine
the minimum $\{-1,1\}$-sequence length for when there exist $k$ consecutive
terms that sum to zero. We determine the corresponding minimum sequence length
when the set $\{-1,1\}$ is replaced by $\{-r,s\}$ for arbitrary positive
integers $r$ and $s.$ This confirms a conjecture of theirs. We also construct
$\{-1,1\}$-sequences of length quadratic in $k$ that avoid $k$ terms indexed by
an arithmetic progression that sum to zero. This solves a second conjecture of
theirs in the case of $\{-1,1\}$-sequences on zero-sum arithmetic subsequences.
Finally, we give a superlinear lower bound on the minimum sequence length to
find a zero-sum arithmetic progression for general $\{-r,s\}$-sequences.
