Cell-free massive multiple-input-multiple-output (mMIMO) is an emerging technology for beyond 5G with its promising features such as higher spectral efficiency and superior spatial diversity as compared to conventional multiple-input-multiple-output (MIMO) technology. The main working principle of cell-free mMIMO is that many distributed access points (APs) cooperate simultaneously to serve all the users within the network without creating cell boundaries. This paper considers the uplink of a cell-free mMIMO system utilizing the radio stripe network architecture with a sequential fronthaul between the APs. A novel uplink sequential processing algorithm is developed, which is proved to be optimal in both the maximum spectral efficiency (SE) and the minimum mean square error (MSE) sense. A detailed quantitative analysis of the fronthaul requirement or signaling of the proposed algorithm and its comparison with competing sub-optimal algorithms is provided. Key conclusions and implications are summarized in the form of corollaries. Based on the analytical and numerical simulation results, we conclude that the proposed scheme can significantly reduce the fronthaul signaling, without compromising the communication performance.