The sum of #'s 1 through 10 = 55 What mathmatical theory or "lesson" is this, and why does it work?

I randomly remembered how to do this the other day, and it freaked me out... If you take the first number and add it to the 2nd number, find the average between the two, and then multiply the last # in the series by the average, it will give you the sum of the #'s in the series. I feel like I know the buttons on a calculator that do this, but couldn't do it after not being in school for so long, but I really want to know the name of this, as well as why this works.

For the visual people out there... the sum of #'s 1-10 = ?

1 + 10 = 11

11/2 = 5.5

5.5 * 10 = 55, which is the sum of the numbers in that series.

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  • 1 decade ago
    Best Answer

    This is called the Gaussian summing formula, and is believed to have been discovered by Gauss as a boy. (This is the same man that later became an excellent physicist who revolutionized our understanding of magnetic fields).

    This works because there are paired numbers at each end that add to the same. Since there are two numbers in a pair, you find this sum, then use half the count.

    For the numbers from 1 to 10. Gauss realized :

    1 + 10 = 11

    and so does 2 + 9

    and also 3 + 8

    and then 4 + 7

    and finally 5 + 6

    There are 5 pairs totalling 11 each, for a total of 55.

    The actual formula has you add the first and the last numbers, multiply by how many numbers there are, then divide by two.

    So for the sum of the numbers from 1 to 10 you would do :

    1 + 10 = 11 : the sum of the first and last number

    10 / 2 = 5 : there are 10 numbers, so use half the count, since we are counting pairs of numbers

    11 * 5 = 55 : then multiply for the answer

    and that is the sum of the numbers from 1 to 10.

    Note that your way of doing this will not work unless the sum begins with 1, or is a natural count of consecutive numbers. If the numbers start with 2, or are by counts of 2, your system fails you.

    The Gaussian way to find the sum of all the even numbers from 2 to 22 is :

    2 + 22 = 24

    11 numbers in list

    24/2 = 12 : half the sum for the pairs

    12 * 11 = 132 : times the count for the total

    If you used your form of this, you will end up with the false assurance that this totals 264 :

    2 + 22 = 24 : sum the first and last

    24 / 2 = 12 : half the sum

    then take this times 22 (The last number in the series that you thought you used)

    and end up wrong with 12 * 22 = 264

    The multiplier is how many numbers in the list -- and there are only 11 numbers on the list of even numbers from 2 to 22.

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