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Thread: Binomial expansion

  1. #1
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    Binomial expansion

    Hi guys!

    I have a homework question about the binomial expansion:

    The first four terms, in ascending powers of x, of the binomial expansion of (1+kx)^n are
    1 + Ax + Bx^2 + Bx^3 + ...
    where k is a positive constant and A, B and n are positive integers.

    a) By considering the coefficients of x^2 and x^3, show that 3=(n-2)k

    Any help would be really great

    Thanks in advance!
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  2. #2
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    $\displaystyle \left( {1 + kx} \right)^n = \sum\limits_{j = 0}^n {\binom{n}{j}\left({kx} \right)^j } = 1 + nkx + \frac{{n\left( {n - 1} \right)}}
    {{2!}}k^2 x^2 + \frac{{n(n - 1)(n - 2)}}
    {{3!}}k^3 x^3 + \cdots $
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  3. #3
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    I have that part, but i'm not sure how it simplifies down to give 3=(n-2)k
    (this is probably a stupid question...)
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  4. #4
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    $\displaystyle \begin{gathered}
    B = B \hfill \\
    \frac{{n\left( {n - 1} \right)}}
    {{2!}}k^2 = \frac{{n(n - 1)(n - 2)}}
    {{3!}}k^3 \hfill \\
    3 = \left( {n - 2} \right)k \hfill \\
    \end{gathered} $
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