  We consider the space-time-matter theory (STM) in a five-dimensional vacuum
space-time with a generalized FLRW metric to investigate the late-time
acceleration of the universe. For this purpose, we derive the four-dimensional
induced field equations and obtain the evolution of the state parameter with
respect to the redshift. Then, we show that with consideration of the extra
dimension scale factor to be a linear function of redshift, this leads to a
model which gives an accelerating phase in the universe. Moreover, we derive
the geodesic deviation equation in the STM theory to study the relative
acceleration of the parallel geodesics of this space-time, and also, obtain the
observer area-distance as a measurable quantity to compare this theory with two
other models.
