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JCECE Engineering JCECE Engineering Solved Paper-2012

  • question_answer
    Find the value of \[^{1}{{P}_{1}}+2{{\cdot }^{2}}{{P}_{2}}+3{{\cdot }^{3}}{{P}_{3}}+4{{\cdot }^{4}}{{P}_{4}}+...{{+}^{n}}{{P}_{n}}\]

    A) \[^{n+1}{{P}_{n+1}}\]                   

    B) \[^{n+1}{{P}_{n+1}}-1\]

    C) \[^{n+1}{{P}_{n+1}}-2\]                               

    D) \[^{n+1}{{P}_{n+1}}+1\]

    Correct Answer: B

    Solution :

    \[r\text{th}\] term of the given series                 \[r{{\cdot }^{r}}{{p}_{r}}=r\cdot r!=\{(r+1)-1\}r!\]                 \[=(r+1)!-r!\] On putting \[r=1,\,\,2,\,\,3,\,\,...n\] and adding, we get                 \[^{1}{{p}_{1}}+2{{\cdot }^{2}}{{p}_{2}}+3{{\cdot }^{3}}{{p}_{3}}+...+n{{\cdot }^{n}}{{p}_{n}}\]                 \[=(2!\,\,-1!)+(3!\,\,-2!)+(4!\,\,-3!)+...+\]                                                                 \[\{(n+1)!\,\,-n\}\]                 \[=(n+1)!\,\,-1!\]                 \[{{=}^{n+1}}{{P}_{n+1}}-1\]


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