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JEE Main & Advanced Sample Paper JEE Main - Mock Test - 11

  • question_answer
    Let \[<{{a}_{n}}>\] be an arithmetic sequence such that arithmetic mean of \[{{a}_{1}},{{a}_{3}},{{a}_{5}},.....{{a}_{97}},{{a}_{99}}\]is 1. Then the value of  \[\left| \sum\limits_{r=1}^{50}{{{(-1)}^{\frac{r(r+1)}{2}}}.{{a}_{2r-1}}} \right|\] is

    A) \[1\]                 

    B) \[2\]

    C) \[3\]                 

    D) \[4\]

    Correct Answer: B

    Solution :

    [b] Given that \[{{a}_{1}}+{{a}_{3}}+....+{{a}_{99}}=50\] \[\Rightarrow \,\,\,\,({{a}_{1}}+{{a}_{99}})+({{a}_{3}}+{{a}_{97}})+....+({{a}_{47}}+{{a}_{49}})=50\] \[\Rightarrow \,\,\,\,2{{a}_{50}}+2{{a}_{50}}+....25\,\,times=50\] \[\Rightarrow \,\,\,{{a}_{50}}=1\] Now, \[\left| \sum\limits_{r=1}^{50}{{{\left( -1 \right)}^{\frac{r(r+1)}{2}}}.{{a}_{2r-1}}} \right|\] \[=|-{{a}_{1}}-{{a}_{3}}+{{a}_{5}}+{{a}_{7}}-{{a}_{9}}-{{a}_{11}}+...+{{a}_{93}}+{{a}_{95}}\] \[-{{a}_{97}}-{{a}_{99}}|\] \[=|-{{a}_{1}}-{{a}_{99}}|\,\,=\,\,|2{{a}_{50}}|\,\,=2\]


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