The Cramér Rao inequality in the sequential case gives a lower bound for thevariance of an unbiased estimator of a parametric function under finite stopping rules.This article shows that when the observations follow a one parameter exponential familyof distributions the bound can be attained for one or all values of the parameter under strictly sequential rules only in a very special case, namely, for the Bernoullidistribution. Some applications of the result to the construction of optimum estimators are also given. Our main result is a generalization of DeGroot's work for the Bernoulli distribution. Moreover, the main result along with Kagan's theorem can be treated as a generalization of Wijsman's work for nonsequential estimators.