By means of approximate numerical solutions obtained from a first-order correction to the prompt-jump approximation, good agreement is found with exact numerical solutions of the kinetics equations. Accuracies of <0.1% are obtainable for iterative time steps of as much as 1 sec, provided the reactor remains below prompt-critical [i.e., k(t) < $1]. The accuracy increases as l/β → 0, i.e., as the prompt-neutron lifetime becomes smaller or as the reactor becomes “faster.” This is true for both fast- and slow-reactivity insertion rates, C. Two methods for handling rapid reactivity insertion rates are discussed. One (Method A) is more applicable for C ≈ 1 → 50 $/sec, and the other (Method B, which effectively shifts the time scale) is more applicable for C ≳ 50 $/sec. In the one delayed-neutron-group approximation, analytic results are presented for arbitrary reactivity insertion rates and comparisons are made with previous methods.