We consider the space-time-matter theory (STM) in a 5D vacuum spacetime with a generalized FLRW metric to investigate the late-time acceleration of the universe. For this purpose, we derive the 4D 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 spacetime, and also, obtain the observer area-distance as a measurable quantity to compare this theory with two other models.
