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Math Help - Binomial Expansions

  1. #1
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    Binomial Expansions

    Q1)Find, in its simplest form, the coefficient of x^r in the expansion, in ascending powers of  x , of  \frac{1}{x-3}

    Q2) Expand  (1+y)^14 as a series of ascending powers of  y up to and including the term in y^3, Simplify the coefficients.

    I got the above part right and derived at the answer of  1 + 14y + 91y^2 + 364y^3

    However, its the following part that i dont get

    In The expansion of  (1 + x + kx^2)^14 , where  k is a constant, the coefficient of  x^3 is zero. By writing  x + kx^2 as  y , or otherwise, find the value of k.

    Any help is appreciated ^^
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  2. #2
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    Quote Originally Posted by teddybear67 View Post
    Q1)Find, in its simplest form, the coefficient of x^r in the expansion, in ascending powers of  x , of  \frac{1}{x-3}

    Q2) Expand  (1+y)^14 as a series of ascending powers of  y up to and including the term in y^3, Simplify the coefficients.

    I got the above part right and derived at the answer of  1 + 14y + 91y^2 + 364y^3

    However, its the following part that i dont get

    In The expansion of  (1 + x + kx^2)^14 , where  k is a constant, the coefficient of  x^3 is zero. By writing  x + kx^2 as  y , or otherwise, find the value of k.

    Any help is appreciated ^^
    hi

    begin with some substitution as hinted by the question

    1+14(x+kx^2)+91(x+kx^2)^2+364(x+kx^2)^3+91(x^2+2kx ^3+k^2x^4)+364(x^3+3kx^4+3x(kx^2)^2+(kx^2)^3)

    Given that the coefficient of x^3 is 0 , so we need to know the coefficient of x^3 only . To expand the whole thing would be tedious , so we expand those parts where x^3 is .

    ..... 91x^2+182kx^3 + ... +364x^3+....

    coefficient of x^3 : 182k+364\Rightarrow k=-2
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  3. #3
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    Thanks!
    But do ya know how to do the top one too? :/
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  4. #4
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    \frac{1}{x- 3}= (x- 3)^{-1}

    Use the "generalized binomial theorem"
    (a+ b)^r= \sum_{i=0}^\infty\begin{pmatrix}r \\ i\end{pmatrix}a^ib^{r-i}
    Where, for r not a positive integer,
    \begin{pmatrix}r \\ i\end{pmatrix} =  \frac{r(r-1)\cdot\cdot\cdot(r- i+1)}{i!}
    With r not a positive integer, r- i+ 1 is never equal to 0 so this is an infinite series.
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