  In this paper, we provide some solvability conditions in terms of ranks for
the existence of a general solution to a system of $k$ Sylvester-type
quaternion matrix equations with $3k+1$ variables
$A_{i}X_{i}+Y_{i}B_{i}+C_{i}Z_{i}D_{i}+F_{i}Z_{i+1}G_{i}=E_{i},~i=\overline{1,k}$.
As applications of this system, we present rank equalities as the necessary and
sufficient conditions for the existence of a general solution to some systems
of quaternion matrix equations
$A_{i}X_{i}+(A_{i}X_{i})_{\phi}+C_{i}Z_{i}(C_{i})_{\phi}+F_{i}Z_{i+1}(F_{i})_{\phi}=E_{i},~i=\overline{1,k}$.
