We can't all be Gauss but, then again, we're not in fourth grade anymore, either. You can sum over any range trivially: average of the endpoints times length of the range. The average of 1 and 365 is 183. There are 365 elements between 1 and 365 (both inclusive). 183 * 365 = 66795. Yay, right answer.

(There is a formula. I have a poor memory for formulas but am good at transforming English into them, so I just remember the English gloss.)

I'm decently impressed that Wolfram was able to spit out something equivalent to the right formula with only a little massaging of the inputs from me:

http://www.wolframalpha.com/input/?i=sum+of+n+from+n%3Da+to+...