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Math Help - Proof by Mathematical Induction

  1. #1
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    Proof by Mathematical Induction

    Prove that for all integers n >= 1,

    1/3 = (1 + 3)/(5 + 7) = (1 + 3 + 5)/(7 + 9 + 11) = ...
    = (1 + 3 + ... + (2n - 1))/((2n + 1) + ... + (4n - 1))

    This is what I have so far:
    Show that P(1) is true. Left-hand side = 1/3. Right-hand side = 1/3. Since the left-hand side is equal to the right-hand side, P(1) is true. Show that for any integer k >= 1, if P(k) is true, then P(k + 1) is also true.
    Suppose that k is any integer with k >= 1.

    I am confused about how to complete the inductive step. Can anyone help me with this?
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  2. #2
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    Re: Proof by Mathematical Induction

    Quote Originally Posted by lovesmath View Post
    Prove that for all integers n >= 1,
    1/3 = (1 + 3)/(5 + 7) = (1 + 3 + 5)/(7 + 9 + 11) = ...
    = (1 + 3 + ... + (2n - 1))/((2n + 1) + ... + (4n - 1))
    I would do it in two parts.
    The numerator \sum\limits_{n = 1}^N {\left( {2n - 1} \right)}  = N^2 .

    The denominator \sum\limits_{n = N}^{2N-1} {\left( {2n + 1} \right)}  = 3 N^2

    The separate inductions should be easier.
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