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Math Help - Square route question

  1. #1
    Senior Member Mukilab's Avatar
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    Square route question

    Find the pattern or general rule between \sqrt{2*3*4*5+1} and \sqrt{0*1*2*3+1} and \sqrt{3*4*5*6+1}

    I can't see any pattern between them, let alone working out a general rule. Can somebody give me a hint, not the full answer, just a flavour please
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  2. #2
    MHF Contributor undefined's Avatar
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    Quote Originally Posted by Mukilab View Post
    Find the pattern or general rule between \sqrt{2*3*4*5+1} and \sqrt{0*1*2*3+1} and \sqrt{3*4*5*6+1}

    I can't see any pattern between them, let alone working out a general rule. Can somebody give me a hint, not the full answer, just a flavour please
    List them out...
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  3. #3
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    Hello, Mukilab!

    Find the pattern or general rule for: . \begin{Bmatrix} \sqrt{0\!\cdot\!1\!\cdot\!2\!\cdot\!3+1} \\ \sqrt{1\!\cdot\!2\!\cdot\!3\!\cdot\!4+1} \\ \sqrt{2\!\cdot\!3\!\cdot\!4\!\cdot\!5+1} \\ \sqrt{3\!\cdot\!4\!\cdot\!5\!\cdot\!6+1} \end{Bmatrix}

    We have:

    . . \begin{array}{ccccccc}\sqrt{0\!\cdot\!1\!\cdot\!2\  !\cdot\!3 + 1} &=^& \sqrt{1} &=& 1 \\<br />
\sqrt{1\!\cdot\!2\!\cdot\!3\!\cdot\!4+1} &=& \sqrt{25} &=& 5 \\<br />
\sqrt{2\!\cdot\!3\!\cdot\!4\!\cdot\!5+1} &=& \sqrt{121} &=& 11 \\<br />
\sqrt{3\!\cdot\!4\!\cdot\!5\!\cdot\!6+1} &=& \sqrt{361} &=& 19 \\<br />
\sqrt{4\!\cdot\!5\!\cdot\!6\!\cdot\!67 + 1} &=& \sqrt{841} &=& 29 \\ \vdots && \vdots \end{array}


    The general term is:

    . . a_n \;=\;\sqrt{n(n+1)(n+2)(n+3)+1}\;\text{ for } n = 0,1,2,\hdots

    . . . . . =\; \sqrt{n^4 + 6n^3 + 11n^2 + 6n + 1}

    . . . . . =\;\sqrt{(n^2 + 3n + 1)^2}

    . . . . . =\; n^2+3n+1

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