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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 the expansion of \[{{(1+x)}^{n}}\]the coefficient of pth and \[{{(p+1)}^{th}}\] terms are respectively p and q. Then \[p+q=\] [EAMCET 2002]

    A) \[n+3\]

    B) \[n+1\]

    C) \[n+2\]

    D) \[n\]

    Correct Answer: B

    Solution :

    pth term \[={{T}_{p}}={}^{n}{{C}_{p-1}}{{(x)}^{n-p+1}}{{(1)}^{p-1}}=p\] (p + 1)th term \[={{T}_{p+1}}={}^{n}{{C}_{p}}{{(x)}^{n-p}}{{(1)}^{p}}=q\] Then, coefficient of \[\frac{p}{q}=\frac{{}^{n}{{C}_{p-1}}}{{}^{n}{{C}_{p}}}\] Þ \[\frac{p}{q}=\frac{n!}{\left( p-1 \right)\,!\,\left( n-p+1 \right)\,\,!}\,.\,\frac{p\,!\,\,\,\left( n-p \right)\,\,!}{n\,!}\] Þ \[\frac{p}{q}=\frac{p}{n-p+1}\] Þ  \[p+q=n+1\].


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