We present a study of perpendicular magnetic anisotropy in [Co/Ni(t)/Co/Pt]×8 multilayers for use as free layers in magnetic tunnel junctions (MTJ) and spin valves. The thickness t of the Ni sub-layer was varied and the resulting magnetic properties were investigated. As determined from magnetic force microscopy and magnetometry measurements, all multilayers exhibited a perpendicular magnetic anisotropy with an increase of saturation magnetization with thickness t. From the temperature dependence of the magnetization, well described by a Bloch law, we find that the spin-wave stiffness constant of the [Co/Ni(t)/Co/Pt]×8 multilayers is larger compared to (Co/Ni) multilayers. These multilayers could be the basis for spintronic devices where the reduction of total Pt content could help to reduce the damping constant while keeping the magnetic anisotropy energy relatively high. These are conflicting requirements needed for high performance magnetic memory devices.