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Math Help - [SOLVED] Proof by induction help needed (urgently)

  1. #1
    suf
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    [SOLVED] Proof by induction help needed (urgently)

    This is question 4 on June 2008 Further Pure 1 paper OCR (downloadable for OCR website) OCR > Qualifications > By type > AS/A Level GCE (current) > Mathematics > Mathematics > All documents

    The question goes:
    The Matrix A is given by A =

    3, 1,
    0, 1,

    prove by induction that for n>1

    A^n =

    3^n, 0.5(3^n -1),
    0, 1,

    I just dont get how 3^k + 0.5(3^k -1) becomes 0.5(3^k+1 -1)

    Thanks
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  2. #2
    Junior Member slider142's Avatar
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    Quote Originally Posted by suf View Post
    ...
    I just dont get how 3^k + 0.5(3^k -1) becomes 0.5(3^k+1 -1)

    Thanks
    Simplify the expression: 3^k + \frac{1}{2} 3^k - \frac{1}{2}
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  3. #3
    suf
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    Thats what im having probelms with
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  4. #4
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    Quote Originally Posted by slider142 View Post
    Simplify the expression: 3^k + \frac{1}{2} 3^k - \frac{1}{2}
    3^k + \frac{1}{2}\cdot 3^k - \frac{1}{2} = \left(1 + \frac{1}{2}\right)3^k - \frac{1}{2}

    = \frac{3}{2}\cdot 3^k - \frac{1}{2}

     = \frac{1}{2}\cdot 3^{k + 1} - \frac{1}{2}

     = \frac{1}{2}(3^k - 1).
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