The geometric process (GP) plays an important role in the reliability theory and life span models. It has been used extensively as a stochastic model in many areas of application. Therefore, the parameter estimation problem is very crucial in a GP. In this study, the parameter estimation problem for GP is discussed under the assumption that X1 has a Lindley distribution with parameter theta. The maximum likelihood (ML) estimators of a,mu and sigma(2) of the GP and their asymptotic distributions are derived. A test statistic is developed based on ML estimators for testing whether a=1 or not. The same problem is also studied by using Bayesian methods. Bayes estimators of the unknown model parameters are obtained under squared error loss function (SELF) using uniform and gamma priors on the ratio a and theta parameters. It is not possible to obtain Bayes estimators in explicit forms. Therefore, Markov Chain Monte Carlo (MCMC), Lindley (LD), and Tierney-Kadane (T-K) methods are used to estimate the parameters a,mu and sigma(2) in GP. The efficiencies of the ML estimators are compared with Bayes estimators via an extensive Monte Carlo simulation study. It is seen that the Bayes estimators perform better than the ML estimators. Two real-life examples are also presented for application purposes. The first data set concerns the coal mining disaster. The second is the number of COVID-19 patients in Turkey.