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JEE Main & Advanced Mathematics Sequence & Series Question Bank Arithmetic Progression

  • question_answer
    The first term of an A.P. of consecutive integers is \[{{p}^{2}}+1\] The sum of \[(2p+1)\] terms of this series can be expressed as

    A) \[{{(p+1)}^{2}}\]

    B) \[{{(p+1)}^{3}}\]

    C) \[(2p+1){{(p+1)}^{2}}\]

    D) \[{{p}^{3}}+{{(p+1)}^{3}}\]

    Correct Answer: D

    Solution :

    \[{{S}_{2p+1}}=\frac{2p+1}{2}\{2({{p}^{2}}+1)+(2p+1-1)\,1\}\] \[=\left( \frac{2p+1}{2} \right)\,(2{{p}^{2}}+2p+2)=(2p+1)({{p}^{2}}+p+1)\] \[={{p}^{3}}+{{(p+1)}^{3}}\].


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