JEE Main & Advanced Mathematics Binomial Theorem and Mathematical Induction Question Bank Properties of binomial coefficients

  • question_answer
    If \[{{(1+x)}^{n}}={{C}_{0}}+{{C}_{1}}x+{{C}_{2}}{{x}^{2}}+....+{{C}_{n}}{{x}^{n}}\], then the value of \[{{C}_{0}}+2{{C}_{1}}+3{{C}_{2}}+....+(n+1){{C}_{n}}\]will be [MP PET 1996; RPET 1997; DCE 1995; AMU 1995; EAMCET 2001; IIT 1971]

    A)   \[(n+2){{2}^{n-1}}\]

    B)   \[(n+1){{2}^{n}}\]

    C)   \[(n+1){{2}^{n-1}}\]

    D)   \[(n+2){{2}^{n}}\]

    Correct Answer: A

    Solution :

    Trick: Put n=1, the expression is equivalent to \[^{1}{{C}_{0}}+{{2.}^{1}}{{C}_{1}}=1+2=3\] Only option (a) gives the value.


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