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Math Help - Math Contest Question 4

  1. #1
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    Math Contest Question 4

    This problem was from the 2004 Luzerne County Math Contest:
    If (r + 1/r)^2 = 3, what is r + (1/r^3) = ?

    Any help would really be appreciated. Thanks in advance!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by eric1299171 View Post
    This problem was from the 2004 Luzerne County Math Contest:
    If (r + 1/r)^2 = 3, what is r + (1/r^3) = ?

    Any help would really be appreciated. Thanks in advance!
    I don't know if there is a more elegant way of solving this but:

    You can solve

    \displaystyle \left( r+\frac{1}{r}\right)^2=3

    which has complex roots but which can be represented in exponential form quite nicely.

    Substitute into:

    \displaystyle r+\left(\frac{1}{r}\right)^3

    and simplify, and if what I have done has no algebraic errors you should end up with r=\pm i

    (note: in a competition you will get no marks for the final answer but just from the method you used to find it)

    CB
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  3. #3
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    I don't know if this is quite what is wanted but one can show that if \left(r+ \frac{1}{r}\right)^2= 3, then r+ \frac{1}{r^3}= \frac{1}{r}.
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  4. #4
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    Hello, eric1299171!

    Is there a typo?
    If the second expression has \,r^3 twice,
    . . there is a rather elegant solution.
    And it looks more like a contest problem.


    \text{Given: }\:\left(r + \dfrac{1}{r}\right)^2 \,=\, 3.
    . . . . . . . .
    \text{Find: }\:r^3 + \dfrac{1}{r^3}

    We have: . . . . . r + \dfrac{1}{r} \;=\;\sqrt{3}

    Then: . . . . . \left(r + \dfrac{1}{r}\right)^3 \;=\;3\sqrt{3}

    . . . . r^3 + 3r + \dfrac{3}{r} + \dfrac{1}{r^3} \;=\;3\sqrt{3}

    . . \displaystyle r^3 + 3\underbrace{\left(r + \frac{1}{r}\right)}_{\text{This is }\sqrt{3}} + \frac{1}{r^3} \;=\;3\sqrt{3}

    . . . . . . \displaystyle r^3 + 3\sqrt{3} + \frac{1}{r^3} \;=\;3\sqrt{3}

    Therefore: . . . . r^3 + \dfrac{1}{r^3} \;=\;0

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  5. #5
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    I checked over the contest and the second expression is definetly r + 1/(r^3). But the answer key has zero. There might be a typo on a contest. Thanks for the reply! =)
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  6. #6
    Super Member TheChaz's Avatar
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    Quote Originally Posted by eric1299171 View Post
    ... Thanks for the reply! =)
    There's a button for that
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