In response to the original post, you don't need to be a mathematician to know this. I heard about this formula in elementary school, with an anecdote supposedly involving Carl Friedrich Gauss and a nasty math teacher.

Regarding the "programmer/mathematician" comparison, be careful, because sometimes you see naive "mathematicians" that:

• suggest a wonderful but extremely complicated O(N) algorithm, without noticing that for a given aplication N is always between 0 and 2, and considering the overhead, the N^2 method is actually faster.
• don't care about the implementation issues of the language they are using, so they use a theoretically pretty but unnecessary recursive implementation that ends up being too slow or resource-intensive.

The point that thinking is sometimes better than brute force is well taken, but sometimes the opposite is good enough or better. From the Jargon File:

Ken Thompson, co-inventor of Unix, is reported to have uttered the epigram "When in doubt, use brute force". He probably intended this as a ha ha only serious, but the original Unix kernel's preference for simple, robust, and portable algorithms over brittle 'smart' ones does seem to have been a significant factor in the success of that OS. Like so many other tradeoffs in software design, the choice between brute force and complex, finely-tuned cleverness is often a difficult one that requires both engineering savvy and delicate esthetic judgment.