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Math Help - multiple of 4

  1. #1
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    multiple of 4

    Each of the numbers

    a_1,a_2,...,a_n

    is either 1 or -1.

    If the sum


    S =a_1a_2a_3a_4+ a_2a_3a_4a_5+ a_3a_4a_5a_6 +.... + a_na_1a_2a_3= 0,


    prove that n

    must be a multiple of 4.
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  2. #2
    Super Member
    Joined
    May 2009
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    Quote Originally Posted by perash View Post
    Each of the numbers
    a_1,a_2,...,a_n
    is either 1 or -1.
    If the sum
    S =a_1a_2a_3a_4+ a_2a_3a_4a_5+ a_3a_4a_5a_6 +.... + a_na_1a_2a_3= 0,
    prove that n
    must be a multiple of 4.
    If each of the numbers a_1,a_2,...,a_n is either a 1 or a -1, then each of the products in the sum above must be 1 or -1. In order for the entire sum to be 0,
    (1) the number of these products added must be even, meaning that n must be even, and
    (2) half of these products (n/2) equal 1 and half of these products (n/2) equal -1.

    Now, instead of taking the sum of these products, let's take the product of these products. This product would be
    P = (a_1a_2a_3a_4)(a_2a_3a_4a_5)(a_3a_4a_5a_6)....(a_n  a_1a_2a_3) = (1)^{n/2}(-1)^{n/2} = (-1)^{n/2}.

    On the other hand, each of a_1,a_2,...,a_n appears 4 times, so (-1)^{n/2} = 1. This means that n/2 must be even, which means that n must be a multiple of 4.


    01
    Last edited by yeongil; July 25th 2009 at 09:36 AM.
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