Summation

Legend has it that Descartes’ teacher asked him to add up the numbers from 1 to 100 to give him something to do that would take some time.

In the summation below, the index i goes from 1 to 100:

$\displaystyle \sum_{i=1}^{100} i = 1 +2 + 3 + ... + 99 + 100$

Back to the story about Descartes. He answered the question much quicker than expected, because he pulled a trick:

1 matched with 99 gives 100, 2 matched with 98 gives 100, and this is true for 1 through 49 — we get 4,900 from these matches
50 wasn’t included in the matches, the last match was 49+51, so add 50
100 wasn’t included in the matches, so add 100

4,900 + 50 + 100 = 5050