Add all integers 1 to 500?

Answer 1

There's a trick to this one. Let 1 + 2 + ... + 500 be the sum of all integers 1 to 500, called X

Now, imagine we want to add the sum the integers from 1 to 500 and to the sum of the integers from 1 to 500. This would give us 2*X

we can write it as

1 + 2 + ... + 500

+

1 + 2 + ... + 500

But that's not particularly useful. What if we look at it as:

1 + 2 + ... + 499 + 500

+

500 + 499 + ... + 2 + 1

adding the numbers that are directly on top of each other, we get

500 + 1 = 501, 499 + 2 = 501... 1 + 500 = 501

Thus every term is 501, and we have 500 terms. So, we have 501*500 = 2X

Thus X, the sum from 1 to 500, is (500*501)/2 = 125,250