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Math Help - 100 Square problem

  1. #1
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    100 Square problem

    Hi, I'm wondering if someone can help with a problem I have.

    If you have a hundred square, 10x10 with numbers 1 - 10 in the first row etc., and you have 2x2, 3x3 squares etc. starting from 1 (top left), is there an equatuion for working out the sum total of the numbers within the square, so I could work out the total of the numbers within an nxn square?

    Thanks in advance.
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  2. #2
    MHF Contributor
    Grandad's Avatar
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    Hello Cornopolis

    Welcome to Math Help Forum!
    Quote Originally Posted by Cornopolis View Post
    Hi, I'm wondering if someone can help with a problem I have.

    If you have a hundred square, 10x10 with numbers 1 - 10 in the first row etc., and you have 2x2, 3x3 squares etc. starting from 1 (top left), is there an equatuion for working out the sum total of the numbers within the square, so I could work out the total of the numbers within an nxn square?

    Thanks in advance.
    On row 1, the sum is:
    1+2+..+n=\tfrac12n(n+1)
    On row 2:
    (10+1)+(10+2)+...+(10+n) = 10n + \tfrac12n(n+1)
    On row 3:
    (2\times10+1)+(2\times10+2)+...+(2\times10+n) = 2\times10n + \tfrac12n(n+1)
    ...

    On row n:
    ([n-1]\times10+1)+([n-1]\times10+2)+...+([n-1]\times10+n) = [n-1]\times10n + \tfrac12n(n+1)
    So the total is:
    (0+1+ 2 + [n-1])10n+n\times\tfrac12n(n+1)
    =\tfrac12(n-1)n\times10n +\tfrac12n^2(n+1)

    =\tfrac12n^2(10n-10+n+1)

    =\tfrac12n^2(11n-9)
    Grandad
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  3. #3
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    Solved

    Brilliant, thanks very much.
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