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JEE Main & Advanced Mathematics Binomial Theorem and Mathematical Induction Question Bank Multinomial theorem, Terms free from radical sign in the expansion(a1/p+b1/q), Problems regarding to three four consecutive terms or coefficients

  • question_answer
     If \[{{a}_{r}}\] is the coefficient of \[{{x}^{r}}\], in the expansion of \[{{(1+x+{{x}^{2}})}^{n}}\], then \[{{a}_{1}}-2{{a}_{2}}+3{{a}_{3}}-....-2n\,{{a}_{2n}}=\] [EAMCET 2003]

    A) 0

    B) n

    C) ? n

    D) 2n

    Correct Answer: C

    Solution :

    Let us take \[{{a}_{0}}+{{a}_{1}}x+{{a}_{2}}{{x}^{2}}+.....+{{a}_{2n}}{{x}^{2n}}={{(1+x+{{x}^{2}})}^{n}}\] Differentiating with respect to x on both sides \[{{a}_{1}}+2{{a}_{2}}x+...+2n\,{{a}_{2n}}{{x}^{2n-1}}\] = \[n{{(1+x+{{x}^{2}})}^{n-1}}(2x+1)\] Put x = ? 1Þ \[{{a}_{1}}-2{{a}_{2}}+3{{a}_{3}}-....+2n\,{{a}_{2n}}=-n\].


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