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JEE Main & Advanced Mathematics Binomial Theorem and Mathematical Induction Question Bank General term, Coefficient of any power of x, Independent term, Middle term and Greatest term and Greatest coefficient

  • question_answer
    If the coefficients of \[{{x}^{7}}\] and \[{{x}^{8}}\]in \[{{\left( 2+\frac{x}{3} \right)}^{n}}\]are equal, then n is [EAMCET 1983; Kurukshetra CEE 1998; DCE 2000; RPET 2001; UPSEAT 2001]

    A) 56

    B) 55

    C) 45

    D) 15

    Correct Answer: B

    Solution :

     \[{{x}^{7}}\],\[{{x}^{8}}\]will occur in \[{{T}_{8}}\]and\[{{T}_{9}}\]. Coefficients of \[{{T}_{8}}\] and \[{{T}_{9}}\]are equal. \[\therefore {{\,}^{n}}{{C}_{7}}{{2}^{n-7}}{{\left( \frac{1}{3} \right)}^{7}}={{\,}^{n}}{{C}_{8}}{{2}^{n-8}}{{\left( \frac{1}{3} \right)}^{8}}\,\,\Rightarrow n=55\].


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