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Thread: Mathematical Induction

  1. #1
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    Mathematical Induction

    Hello.. Please help me to solve this question. I had tried to solve it but then I not get the same result if P(k+1) inserted in the question.
    Thanks for help me.

    Mathematical Induction-111.jpg

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  2. #2
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    Re: Mathematical Induction

    anyone know the answer? help me please.
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  3. #3
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    Re: Mathematical Induction

    The equation has a typo on the right hand side - it should be:

     \prod\limits _{r=1}^n (1+x^{2^r}) = \frac {1-x^{2^{n+1}}}{1-x^2}

    (note that the exponent in the numerator on the right hand side has 2^(n+1) in it, not 2^(r+1).

    Using induction you first prove that this is true for n=1. Then show that if it's true for n=k it's also true for n = k+1.

    Given:  \prod\limits _{r=1}^n (1+x^{2^r}) = \frac {1-x^{2^{n+1}}}{1-x^2} , then

     \prod\limits _{r=1}^{k+1} (1+x^{2^r}) = \left( \prod\limits _{r=1} ^{k}  (1+x^{2^r}) \right) (1+x^{2^{k+1}}) = \frac  {1-x^{2^{k+1}}}{1-x^2} (1+x^{2^{k+1}})

    which equals  \frac {1-x^{2^{k+2}}}{1-x^2}
    Last edited by ChipB; Mar 22nd 2017 at 10:48 AM.
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  4. #4
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    Re: Mathematical Induction

    thanks for the response. I really appreciate it.
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