There is a well known story about Karl Friedrich Gauss when he was in
elementary school. His teacher got mad at the class and told them to
add the numbers 1 to 100 and give him the answer by the end of the
class. About 30 seconds later Gauss gave him the answer.
The other kids were adding the numbers like this:
1 + 2 + 3 + . . . . + 99 + 100 = ?
But Gauss rearranged the numbers to add them like this:
(1 + 100) + (2 + 99) + (3 + 98) + . . . . + (50 + 51) = ?
If you notice every pair of numbers adds up to 101. There are 50
pairs of numbers, so the answer is 50*101 = 5050. Of course Gauss
came up with the answer about 20 times faster than the other kids.
In general to find the sum of all the numbers from 1 to N:
1 + 2 + 3 + 4 + . . . . + N = (1 + N)*(N/2)
That is "1 plus N quantity times N divided by 2."
Hope this helps.
