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Math Help - Algebraic Proof

  1. #1
    Junior Member SuperCalculus's Avatar
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    Algebraic Proof

    So yeah, basically I got my GCSE Maths result back, and I only dropped one mark, but I'm not sure how or where I lost it. I'll post my method, and then you guys can tell me what I did wrong.

    Question: Prove that the sum of two consecutive odd integers is always even.

    My method;

    Let two consecutive odd integers be x and x + 2.

    Hence, x + (x + 2) = Even?

    2x + 2 = Even

    Since twice anything is always even;

    Even + even = even, which is true. Therefore, the sum of two consecutive odd integers is always even. Any idea where I dropped the mark? I only got two of a possible three.
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  2. #2
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    Quote Originally Posted by SuperCalculus View Post
    So yeah, basically I got my GCSE Maths result back, and I only dropped one mark, but I'm not sure how or where I lost it. I'll post my method, and then you guys can tell me what I did wrong.

    Question: Prove that the sum of two consecutive odd integers is always even.

    My method;

    Let two consecutive odd integers be x and x + 2.

    Hence, x + (x + 2) = Even?

    2x + 2 = Even

    Since twice anything is always even;

    Even + even = even, which is true. Therefore, the sum of two consecutive odd integers is always even. Any idea where I dropped the mark? I only got two of a possible three.
    Prehaps you were expected to consider 2k+1~\&~2k+3.
    That makes clear that the integers are odd.
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  3. #3
    Junior Member SuperCalculus's Avatar
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    Quote Originally Posted by Plato View Post
    Prehaps you were expected to consider 2k+1~\&~2k+3.
    That makes clear that the integers are odd.
    Yeah, I'm aware that I proved it for all integers; but that's the point, ALL integers, and therefore including the odd ones, surely? =/.
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  4. #4
    MHF Contributor harish21's Avatar
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    Quote Originally Posted by SuperCalculus View Post
    So yeah, basically I got my GCSE Maths result back, and I only dropped one mark, but I'm not sure how or where I lost it. I'll post my method, and then you guys can tell me what I did wrong.

    Question: Prove that the sum of two consecutive odd integers is always even.

    My method;

    Let two consecutive odd integers be x and x + 2.

    Hence, x + (x + 2) = Even?

    Even + even = even, which is true. Therefore, the sum of two consecutive odd integers is always even. Any idea where I dropped the mark? I only got two of a possible three.
    2x + 2 = Even

    Since twice anything is always even;

    2x+2 = 2(x+1)

    since x+1(=m, suppose) is also an integer. Then 2m is an even number.

    I would have started it this way:

    Let n be any integet. Then 2n+1 is always odd. The next consecutive integer will be 2n+1+2=2n+3
    so the sum is 4n+4= 4(n+1) which is even
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  5. #5
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    Quote Originally Posted by SuperCalculus View Post
    Yeah, I'm aware that I proved it for all integers; but that's the point, ALL integers, and therefore including the odd ones, surely? =/.
    But it did said two consecutive odd integers.
    Some graders can be very picky.
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