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    f(n) ≡ n3 + 7n2 + 14n + 3.

    Express f(n) in the form f(n) ≡ (n + 1)(n + a)(n + b) + c,where a, b and c are integers.

    Answer to this is: f(n) = (n + 1)(n + 2)(n + 4) − 5

    However the next part asks: Hence, show that f(n) is odd for all positive integer values of n.

    The answer is:

    (n + 1) and (n + 2) are consecutive integers
    ∴ either (n + 1) or (n + 2) is even
    ∴ (n + 1)(n + 2)(n + 4) is even
    ∴ (n + 1)(n + 2)(n + 4) − 5 is odd

    Can someone explain this?
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    (Original post by TSRforum)
    f(n) ≡ n3 + 7n2 + 14n + 3.

    Express f(n) in the form f(n) ≡ (n + 1)(n + a)(n + b) + c,where a, b and c are integers.

    Answer to this is: f(n) = (n + 1)(n + 2)(n + 4) − 5

    However the next part asks: Hence, show that f(n) is odd for all positive integer values of n.

    The answer is:

    (n + 1) and (n + 2) are consecutive integers
    ∴ either (n + 1) or (n + 2) is even
    ∴ (n + 1)(n + 2)(n + 4) is even
    ∴ (n + 1)(n + 2)(n + 4) − 5 is odd

    Can someone explain this?
    which part are you unsure about?
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    (Original post by dpm)
    which part are you unsure about?
    I'm unsure about the answer to the second part. It doesn't make sense.
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    (Original post by TSRforum)
    I'm unsure about the answer to the second part. It doesn't make sense.
    but which part of it doesn't make sense?
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    (Original post by TSRforum)
    f(n) ≡ n3 + 7n2 + 14n + 3.

    Express f(n) in the form f(n) ≡ (n + 1)(n + a)(n + b) + c,where a, b and c are integers.

    Answer to this is: f(n) = (n + 1)(n + 2)(n + 4) − 5

    However the next part asks: Hence, show that f(n) is odd for all positive integer values of n.

    The answer is:

    (n + 1) and (n + 2) are consecutive integers
    ∴ either (n + 1) or (n + 2) is even
    ∴ (n + 1)(n + 2)(n + 4) is even
    ∴ (n + 1)(n + 2)(n + 4) − 5 is odd

    Can someone explain this?
    As it says, n+1 and n+2 are two consecutive integers. Out of any pair of consecutive integers 1 must be even since every other integer is even.

    Therefore one of n+1 or n+2 is divisible by 2. Hence (n+1)(n+2)(n+4) must be divisible by 2 hence that is even. An even number minus an odd number is odd hence (n+1)(n+2)(n+4) - 5 must be odd.

    Not sure which part exactly you wanted explaining so I just did all of it.
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    (Original post by 16Characters....)
    As it says, n+1 and n+2 are two consecutive integers. Out of any pair of consecutive integers 1 must be even since every other integer is even.

    Therefore one of n+1 or n+2 is divisible by 2. Hence (n+1)(n+2)(n+4) must be divisible by 2 hence that is even. An even number minus an odd number is odd hence (n+1)(n+2)(n+4) - 5 must be odd.

    Not sure which part exactly you wanted explaining so I just did all of it.
    Thanks! Wouldn't it be easier to write odd x even = odd therefore (n+1)(n+2)(n+4) is even. Even - odd = odd therefore (n+1)(n+2)(n+4) -5 is odd? How would I know which explanation I would get marks for?
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    (Original post by TSRforum)
    Thanks! Wouldn't it be easier to write odd x even = odd therefore (n+1)(n+2)(n+4) is even. Even - odd = odd therefore (n+1)(n+2)(n+4) -5 is odd? How would I know which explanation I would get marks for?
    You would get marks for both. I would stick to your answer to be honest.

    I was just including divisibility by 2 to make it crystal clear that even x anything = even since I wasn't sure which bit you didn't understand.
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    (Original post by TSRforum)
    Thanks! Wouldn't it be easier to write odd x even = odd therefore (n+1)(n+2)(n+4) is even. Even - odd = odd therefore (n+1)(n+2)(n+4) -5 is odd? How would I know which explanation I would get marks for?
    You won't get any marks for writing "odd x even = odd"

    But you should try to get into the habit of expressing yourself as mathematically as possible, so recognizing that n and n+1 are consecutive numbers and hence one must be even (and the equivalent statement for n+1 and n+2) is a useful phrase to know.

    (Similarly, if you have something like n(n+1)(n+2) where n is an integer, you know that one factor must be divisible by 3 so the whole product is too.)
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    (Original post by davros)
    You won't get any marks for writing "odd x even = odd"

    But you should try to get into the habit of expressing yourself as mathematically as possible, so recognizing that n and n+1 are consecutive numbers and hence one must be even (and the equivalent statement for n+1 and n+2) is a useful phrase to know.

    (Similarly, if you have something like n(n+1)(n+2) where n is an integer, you know that one factor must be divisible by 3 so the whole product is too.)
    What are ways are there for expressing yourself as mathematically as possible? Do the exams just expect you to be able to do it?
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    (Original post by TSRforum)
    What are ways are there for expressing yourself as mathematically as possible? Do the exams just expect you to be able to do it?
    It's partly input from your teacher, who should be training you to do this, and partly as a result of practice - the more you do questions like this, the more you will get used to writing structured arguments.

    It's also a good idea to re-read something you've written and ask yourself "am I really convinced by this argument?". Or you could get a friend to do this for you, and do the same for them in return!
 
 
 
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