When calculating the expected value of X in part B, the summation is solved by using the sum of an arithmetic series. In an arithmetic series, each successive term has a constant difference, which in this case is just 1. Here we're summing from 1 to n, which equals n(n+1)/2.
In response to Arie, part B's random variable doesn't really have a typical classification (i.e., not binomial or geometric or anything) since the PMF is just 1/n.
