  In this paper, we are mainly concerned with the blow-up algebras of the
secant varieties of balanced rational normal scrolls. In the first part, we
give implicit defining equations of their associated Rees algebras and fiber
cones. Consequently, we can tell that the fiber cones are Cohen--Macaulay
normal domains. Meanwhile, these fiber cones have rational singularities in
characteristic zero, and are $F$-rational in positive characteristic. The
Gorensteinness of the fiber cones can also be characterized. In the second
part, we compute the Castelnuovo--Mumford regularities and
$\mathbf{a}$-invariants of the fiber cones. We also present the reduction
numbers of the ideals defined by the secant varieties.
