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    So I have this question asking for a proof of a relationship between the second and third co-efficients of the Binomial Expansion:

    (a+x)^n

    are always equal to:

    \frac{(r + 1)a}{n - r}

    The coefficients follow n and \frac{n(n + 1)}{2}

    But how can I pull these together to one simple proof?

    :confused:
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    (Original post by Gmart)
    So I have this question asking for a proof of a relationship between the second and third co-efficients of the Binomial Expansion:

    (a+x)^n

    are always equal to:

    \frac{(r + 1)a}{n - r}

    The coefficients follow n and \frac{n(n + 1)}{2}

    But how can I pull these together to one simple proof?

    :confused:
    \alpha and \beta are not necessarily the second and third terms, they are simply two consecutive terms.
    Spoiler:
    Show
    You know that:
    (a+x)^n=\displaystyle\sum_{r=0}^  n {n \choose r} a^{n-r}x^r
    What can you say about the coefficient of x^r?
    Spoiler:
    Show
    Recall:
    {n \choose r}=\dfrac{n!}{r!(n-r)!}
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    for some reason it won't let me edit the text in the above spoiler.
    It should read:
    {n \choose r}=\dfrac{n!}{r!(n-r)!}.
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    (Original post by joostan)
    \alpha and \beta are not necessarily the second and third terms, they are simply two consecutive terms.
    Seems like a duplicate post to me, I've answered this question in a thread an hour or so ago.
 
 
 
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