Periodic gossip algorithms have generated a lot of interest due to their ability to compute the global statistics by using local pairwise communications among nodes. Simple execution, robustness to topology changes, and distributed nature make these algorithms quite suitable for wireless sensor networks (WSN). However, these algorithms converge to the global statistics after certain rounds of pair-wise communications. A significant challenge for periodic gossip algorithms is difficult to predict the convergence rate for large-scale networks. To facilitate the convergence rate evaluation, we study a one-dimensional lattice network model. In this scenario, to derive the explicit formula for convergence rate, we have to obtain a closed form expression for second largest eigenvalue of perturbed pentadiagonal matrices. In our approach, we derive the explicit expressions of eigenvalues by exploiting the theory of recurrent sequences. Unlike the existing methods in the literature, this is a direct method which avoids the theory of orthogonal polynomials [18]. Finally, we derive the explicit expressions for convergence rate of the average periodic gossip algorithm in one-dimensional WSNs. We analyze the convergence rate by considering the linear weight updating approach and investigate the impact of gossip weights on the convergence rates for a different number of nodes. Further, we also study the effect of link failures on the convergence rate for average periodic gossip algorithms.