Gossip algorithms have received a lot of attention in recent years due to their ability to compute global statistics using local pair-wise communications. Simple execution, robustness to topology changes, and distributed nature make these algorithms more attractive to wireless sensor network (WSN) applications. Periodic gossip algorithm is a special case of gossip algorithm, where neighbouring nodes gossip at every time instant. Convergence rate plays a fundamental role in effectively measuring the performance of periodic gossip algorithms. However, these algorithms are inherently iterative and estimating their convergence rate for large-scale WSNs is a computationally challenging task. Hence, to utilize the periodic gossip algorithms for sensor networks, it is necessary to study the convergence rate with less computational resources. In this work, we model the WSN as a one-dimensional lattice network and derive the closed form expressions of convergence rate for even and odd number of nodes. To realize the closed-form expressions of convergence rate, we obtain the exact formulas of eigenvalues for pentadiagonal matrices. The numerical analysis reveals the new design insights into the effect of gossip weight, network size, and probability of link failures on the performance of convergence rate. Specifically, we prove that w =0.9 achieves fast convergence rate over w =0.5 for large-scale WSNs. Our theoretical results provide the basic analytical tools for controlling the performance of periodic gossip algorithms in large-scale WSNs.