Insecticide application is a major method of pest control in agriculture. Motivated by this, we develop a stage-structured mathematical model for population dynamics coupled with genetic evolution, in which the population is divided into developmental stages and the frequency of a resistant allele is tracked within each stage over time. Using this model, we study how repeated insecticide applications influence resistance evolution in pest populations. The baseline model combines continuous population dynamics, describing recruitment, maturation, and mortality, with allele-frequency dynamics for the resistant allele and discrete pulse updates representing repeated insecticide exposure. We establish basic qualitative properties of the model, including existence and uniqueness of solutions, positivity and invariance of the biologically relevant region, and we characterize how pulse events affect resistant-allele frequencies. The model is then extended to incorporate fitness trade-offs associated with resistance through genotype-dependent mortality and fecundity. Finally, we apply the model to the life history of an invasive fruit fly, Drosophila suzukii. Numerical solutions are obtained using parameter values informed by empirical datasets from the literature for Drosophila suzukii.