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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?

    Completely stuck - just a hint required, am I on the right tracks with looking at series for the Pascal's Triangle sequences?

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    Just a point: They aren't asking to prove a relationship between the second and third coefficients. They're asking to prove it between the r^{th} and (r+1)^{th}.
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    Does that mean induction?
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    (Original post by Gmart)
    Does that mean induction?
    that looks promising.
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    (Original post by Gmart)
    Does that mean induction?
    I don't think you need to go that far. You can use the formula for the binomial expansion, I think it has to do with the formula for n \choose r.
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    (Original post by EricPiphany)
    I don't think you need to go that far. You can use the formula for the binomial expansion, I think it has to do with the formula for n \choose r.
    (Original post by Gmart)
    Does that mean induction?
    I concur with this, just use the factorial definition of \displaystyle {n \choose r} = \frac{n!}{r!(n-r)!} (I may have written that down incorrectly, so just check to be sure)
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    So the a vale takes care of itself, but the fraction I have down to:

    \frac{r!(n-r)!}{(n-1)!}

    But what is my next step? I need:

    \frac{r + 1}{n - 1}

    :confused:
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    (Original post by Gmart)
    So the a vale takes care of itself, but the fraction I have down to:

    \frac{r!(n-r)!}{(n-1)!}

    But what is my next step? I need:

    \frac{r + 1}{n - 1}

    :confused:
    Coefficient of rth term:

    \displaystyle \alpha = a^{n-r} \cdot \frac{n!}{r!(n-r)!}

    Coefficient of (r+1)th term:

    \displaystyle \beta = a^{n-r - 1} \cdot \frac{n!}{(r+1)!(n-r-1)!}

    So, what can you say about \dfrac{\alpha}{\beta}?
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    Thank you so much
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    (Original post by Gmart)
    Thank you so much
    Very welcome.
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    (Original post by Zacken)
    Very welcome.
    Which module is this from?


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    (Original post by anoymous1111)
    Which module is this from?


    Posted from TSR Mobile
    This is IB - but it's C1/2 knowledge.
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    (Original post by Zacken)
    Coefficient of rth term:

    \displaystyle \alpha = a^{n-r} \cdot \frac{n!}{r!(n-r)!}

    Coefficient of (r+1)th term:

    \displaystyle \beta = a^{n-r - 1} \cdot \frac{n!}{(r+1)!(n-r-1)!}

    So, what can you say about \dfrac{\alpha}{\beta}?
    Where do I go from here?


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    (Original post by anoymous1111)
    Where do I go from here?


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    Simplify the quotient of factorials.

    \displaystyle \frac{\alpha}{\beta} = \displaystyle \frac{a^{n-r}}{a^{n-r-1}} \cdot \frac{n!}{r!(n-r)!} \cdot \frac{(r+1)!(n-r-1)!}{n!} = \cdots
 
 
 
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