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Math Help - Induction Method

  1. #1
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    Induction Method

    What is the following sum?

    (1/1*2) + (1/2*3) + (1/3*4) + . . . + (1/(n-1)n)

    Experiment, Conjucture the value, and then prove by induciton

    / means division, ex. 1/2=.5
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  2. #2
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    I assume that this is

    \frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3\cdot 4} + \dots + \frac{1}{(n - 1)n}.


    Notice that you can write the k^{\textrm{th}} term as

    \frac{1}{k(k + 1)} = \frac{1}{k} - \frac{1}{k + 1}.


    So that means we can rewrite the sum as

    \frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3\cdot 4} + \dots + \frac{1}{(n - 1)n} = \left(\frac{1}{1} - \frac{1}{2}\right) + \left(\frac{1}{2} - \frac{1}{3}\right) + \left(\frac{1}{3} - \frac{1}{4}\right) + \dots + \left(\frac{1}{n - 1} - \frac{1}{n}\right)

     = 1 - \frac{1}{n}, since all but the first and last term cancel.
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