Question

Find the sum of the first one thousand positive integers. Explain how you arrived at your result. Now explain how to find the sum of the first n positive integers, where n is any positive integer, without adding a long list of positive integers by hand and without using a calculator

Answer 1

the integers form an arithmetic sequence, with

a = 1
d = 1

So, the sum of the 1st 1000 integers is

1000/2 (2+999) = 500500

This is just an illustration of what Gauss figured out as a kid. You can pair up the numbers starting from the ends

1+1000
2+999
3+998
...
500+501

There are 500 pairs, so the sum is 500*1001 = 500500