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Math Help - Find range of positive values of x?

  1. #1
    Super Member fardeen_gen's Avatar
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    Find range of positive values of x?

    In the expansion of (1 + x)^18 if the term with greatest Binomial coefficient is also the numerically greatest term, then range of positive values of x is:

    A) [(10/11),(11/10)]
    B) [(9/10), (10/9)]
    C) [(9/11), (11/9)]
    D) [(8/9), (9/8)]
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  2. #2
    MHF Contributor
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    Binomial Coefficients

    Hello fardeen_gen
    Quote Originally Posted by fardeen_gen View Post
    In the expansion of (1 + x)^18 if the term with greatest Binomial coefficient is also the numerically greatest term, then range of positive values of x is:

    A) [(10/11),(11/10)]
    B) [(9/10), (10/9)]
    C) [(9/11), (11/9)]
    D) [(8/9), (9/8)]
    If n is even, the greatest Binomial coefficient of \binom n r is the one in the middle; i.e. where r = \frac{n}{2} (If n is odd, it's either of the two equal terms in the middle.) So here the greatest coefficient is \binom {18}{9}.

    So, if this term also has the greatest value, then

    \binom{18}{8}x^8<\binom{18}{9}x^9, and

    \binom{18}{10}x^{10}<\binom{18}{9}x^9

    \Rightarrow x>\frac{\binom{18}{8}}{\binom{18}{9}}, and

    \Rightarrow x<\frac{\binom{18}{9}}{\binom{18}{10}}

    \Rightarrow \frac{9}{10}<x<\frac{10}{9}

    So the answer is B.

    Grandad
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