In this paper, we introduce the concept of sequential $\mathcal{I}$-convergence spaces and $\mathcal{I}$-Fréchet-Urysohn space and study their properties. We give a sufficient condition for the product of two sequential $\mathcal{I}$-convergence spaces to be a sequential $\mathcal{I}$-convergence space. Finally, we introduce sequential $\mathcal{I}$-convergence groups and obtain an $\mathcal{I}$-completion of these groups satisfying certain conditions.