Week 2 Blog Prompt

So for those of you who haven't seen (or have forgotten,) here is a repost of the prompt for this week:

Part a.) What is the sum of the first n integers for n=1, 2, 3, 4, 5, 6, 7, 8, 9, and 10?
*Note, that is to say, for n=6 you will simply calculate 1+2+3+4+5+6
Part b.) Is there a pattern as n gets larger? What do you intuitively think the pattern might be? If you don't see one immediately, try a few larger values of n (around 15 to 25.)
Part c.) Do you think it's possible to write a formula for the sum of the first n integers for any n (for instance, if I want to know the sum of the first 1,000,000 natural numbers, but don't want to write all of them out, is there a shorter formula that I can simply plug n=1,000,000 into?) If so, what do you think it might be (or if you're not sure, why do you think there is one?) and if not, why don't you think so?
Part d.) If you think there is a formula, can you justify (or prove) that it is correct? If you don't think there is one, can you prove that there isn't? (It's okay if you can't do this part, but just elaborate a bit.)

Keep in mind that you are not expected to do any research (and in fact, it looks worse if you do.) It will be relatively obvious if you use an outside source or another student's blog post as a basis for your own.