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JEE Main & Advanced Mathematics Binomial Theorem and Mathematical Induction Question Bank Self Evaluation Test - Principle of Mathematical Induction

  • question_answer
    Let \[S(k)=1+3+5...+(2k-1)=3+{{k}^{2}}.\] Then which of the following is true?

    A) Principle of mathematical induction can be used to prove the formula

    B) \[S(k)\Rightarrow S(k+1)\]

    C) \[S(k)\not{\Rightarrow }S(k+1)\]

    D) \[S(1)\] is correct

    Correct Answer: B

    Solution :

    [b] \[S(k)=1+3+5+...+(2k-1)=3+{{k}^{2}}\] \[S(1):1=3+1,\] which is not true \[\because \,\,\,S(1)\] is not true \[\therefore \] P.M.I cannot be applied Let S(k) is true, i.e, \[1+3+5...+(2k-1)=3+{{k}^{2}}\] \[\Rightarrow 1+3+5...+(2k-1)+2k+1\] \[=3+{{k}^{2}}+2k+1=3+{{(k+1)}^{2}}\] \[\therefore S(k)\Rightarrow S(k+1)\]


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