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

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  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


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

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