I'm not sure whether it's the notation that's giving you trouble, or the formula in n that equals the sum. See the attached pdf that I hope will help you with the notation. I'm also including another pdf which I got from Purplemath. It will show you a method by which these formulas may be derived.
I'm not so good at entering this notation either. It's not so hard, but I've been lazy. There's a forum and some guidance elsewhere on this site about how to use latex.
Just had a thought ... that pdf I sent you from Purplemath ... just in case you don't see how to apply the method ...
It's true that she's talking about sequences, not series. Well, think of the series represented by the sigma k as a sequence of partial sums. What are the partial sums? You can count them starting at 0 or at 1. If you start at 0, they're 0, 1, 3, 6, 10, 15, 21 ... but if you start at 1 they're 1, 3, 6, 10, 15, 21 ... now set up the simultaneous equations the way she describes and you'll see you arrive at two different formulas.
Hope this helps.