JEE Main & Advanced Mathematics Sequence & Series Question Bank nth term of special series, Sum to n terms and Infinite number of terms

  • question_answer
    The natural numbers are written as follows                                  \[1\]  x\[\begin{matrix}    2 & 3  \\ \end{matrix}\]\[\begin{matrix}    4 & 5 & 6  \\ \end{matrix}\]\[\begin{matrix}    7 & 8 & 9 & 10  \\    . & . & . & .  \\    . & . & . & .  \\    . & . & . & .  \\ \end{matrix}\] The sum of the numbers in the \[{{n}^{th}}\] row is

    A)   \[\frac{n}{2}({{n}^{2}}-1)\]

    B)   \[\frac{n}{2}({{n}^{2}}+1)\]

    C)   \[\frac{2}{n}({{n}^{2}}+1)\]

    D)   \[\frac{2}{n}({{n}^{2}}-1)\]

    Correct Answer: B

    Solution :

    \[{{T}_{n}}=\frac{1}{2}({{n}^{2}}-n+2),\] this is the first term of two \[{{n}^{th}}\] row. Also, the terms of \[{{n}^{th}}\]row form an A.P. with its first term as \[\frac{1}{2}({{n}^{2}}-n+2)\] and a common difference of 1. Hence the sum of the terms in the \[{{n}^{th}}\] row is  \[p:q:r=(-1\pm \sqrt{3})K:2K:(-1\mp \sqrt{3})K\].


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