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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
    In \[{{\left( \sqrt[3]{2}+\frac{1}{\sqrt[3]{3}} \right)}^{n}}\]  if the ratio of \[{{7}^{th}}\] term from the beginning to the \[{{7}^{th}}\] term from the end is \[\frac{1}{6}\], then  \[n=\]

    A) 7

    B) 8

    C) 9

    D) None of these

    Correct Answer: C

    Solution :

    \[\frac{1}{6}=\frac{^{n}{{C}_{6}}{{({{2}^{1/3}})}^{n-6}}{{({{3}^{-1/3}})}^{6}}}{^{n}{{C}_{n-6}}{{({{2}^{1/3}})}^{6}}{{({{3}^{-1/3}})}^{n-6}}}\]or  \[{{6}^{-1}}={{6}^{-4}}{{.6}^{n/3}}={{6}^{n/3-4}}\] \\[\frac{n}{3}-4=-1\] \[\Rightarrow \] n = 9.


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