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Gauss's Trick - Arithmetic Sums

  1. Mar 28, 2014 #1
    I can't grasp the underlying process on how this is working.

    n/2(f+l) = algorithm sum of all integers
    n= number of all integers
    f= first integer
    l= last integer

    Example: 1, 2, 3, 4
    4/2(1+4)
    2(5) = 10

    I know how to do it, but I don't really understand how to actually do it. Am I just too stupid?

    Why do I need to split the sum of all integers?
    Why am I adding the first + last integer?
    Why when I times them together does it work?
    How did he create the algorithm for this?
     
  2. jcsd
  3. Mar 28, 2014 #2

    SteamKing

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    In order to see how this works, write down the string of integers in two different ways:

    Code (Text):

     1  2  3  4  5  6  7  8  9  10
    10  9  8  7  6  5  4  3  2   1
     
    What do you notice about the sum of each column of numbers?
     
  4. Mar 28, 2014 #3

    AlephZero

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    What Gauss did (according to the usual story) was to pair off the numbers like this.
    Suppose you want to sum the 9 numbers 7 8 9 10 11 12 13 14 15
    7 + 15 = 22
    8 + 14 = 22
    9 + 13 = 22
    10 + 12 = 22
    11 = 22/2
    So the sum = (9/2)(22) = (9/2)(7+15)
     
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