Average of sample means

If random samples, each with n=100 scores, are selected from a normal population with mu=1000 and standard deviation=50, and the mean is calculate for each sample, then the average of all the sample means would be what?

Is the answer just 1000?

Well, naturally It would seem that the means of the sample means would tend towards the population mean. However, If it really is a random sample from a normal distribution, the mean of the sample means will never quite reach exctly 1000, but it will come close. It can be shown that the mean of the sample means is also a random variable that converges on the population mean when the sample size becomes very large.