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Binomial Expansion Question Watch

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    Hey I've spent a while trying to figure out part (b) of this question, but here's the whole question anyway.

    9 a) Determine the binomial expansion of in ascending powers of x, up to and including the term in .

    b) Show that the coefficient of in this expansion is for a rational number whose value is to be determined.

    c) Determine the value of correct to fourteen decimal places.
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    (Original post by MahaMaths)
    Hey I've spent a while trying to figure out part (b) of this question, but here's the whole question anyway.

    9 a) Determine the binomial expansion of in ascending powers of x, up to and including the term in .

    b) Show that the coefficient of in this expansion is for a rational number whose value is to be determined.

    c) Determine the value of correct to fourteen decimal places.
    Hi, can you please post all your working up to the point you got stuck? Thanks.
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    (Original post by MahaMaths)
    Hey I've spent a while trying to figure out part (b) of this question, but here's the whole question anyway.

    9 a) Determine the binomial expansion of in ascending powers of x, up to and including the term in .

    b) Show that the coefficient of in this expansion is for a rational number whose value is to be determined.
    For an expansion (1+x)^{k} the coefficient of x^n is \frac{k(k-1)(k-2)...(k-n+1)}{n!}. See how this can help you.
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    (Original post by RDKGames)
    For an expansion (1+x)^{k} the coefficient of x^n is \frac{k(k-1)(k-2)...(k-n+1)}{n!}. See how this can help you.
    Sorry, I tried to post my attempt at working out yesterday, but it was declined after moderation, for some reason.

    Here it is:

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    Please help. Can someone show me the steps to get to the answer in part (b)?
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    (Original post by MahaMaths)
    Sorry, I tried to post my attempt at working out yesterday, but it was declined after moderation, for some reason.
    The numerator is

    -3 \times -4 \times -5 \times \cdots \times (-3-(n-2)) \times (-3-(n-1))

    Try taking out factors of -1 from this then think about what you can cancel in the fraction.
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    Okay, I've got really close, so thanks for your help so far. I'm struggling to see how I can get rid of the (-1)^n though...
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    (Original post by MahaMaths)
    Okay, I've got really close, so thanks for your help so far. I'm struggling to see how I can get rid of the (-1)^n though...
    (-1) \times 3 \times 4 \times 5 \times 6 \neq -3 \times -4 \times -5 \times -6

    Can you see your mistake and how to correct it?
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    (Original post by Notnek)
    (-1) \times 3 \times 4 \times 5 \neq -3 \times -4 \times -5

    Can you see your mistake and how to correct it?
    Just checking - are you sure you meant to say that?
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    (Original post by ghostwalker)
    Just checking - are you sure you meant to say that?
    Damn I was trying to illustrate something but didn't have enough numbers Thanks.

    MahaMaths I have edited my post above.
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    Thanks for your help @Notnek, ghostwalker and RDKGames. I finally got through the proof so it means you've done a good job (and that edited slip-up amused me).

 
 
 
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