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Math Help - Quick Proof

  1. #1
    Junior Member Misa-Campo's Avatar
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    Quick Proof

    How do i prove that the LHS=RHS? i am really confused because of factorials together with pronumerals

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  2. #2
    MHF Contributor red_dog's Avatar
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    1-\frac{1}{(k+1)!}+\frac{k+1}{(k+2)!}=1-\frac{k+2}{(k+2)!}+\frac{k+1}{(k+2)!}=

    =1+\frac{k+1-k-2}{(k+2)!}=1-\frac{1}{(k+2)!}
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  3. #3
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    earboth's Avatar
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    Quote Originally Posted by Misa-Campo View Post
    How do i prove that the LHS=RHS? i am really confused because of factorials together with pronumerals

    I assume that you know

    (k+2)! = (k+1)! \cdot (k+2)

    The common denominator of the fractions therefore is (k+2)!

    1-\frac1{(k+1)!} + \frac{k+1}{(k+2)!} = 1-\frac{k+2}{(k+1)! \cdot (k+2)} + \frac{k+1}{(k+2)!} \ =\ 1-\left(\frac{k+2}{(k+1)! \cdot (k+2)} - \frac{k+1}{(k+2)!}\right) = 1-\frac{(k+2)-(k+1)}{(k+2)!}

    which will yield the RHS.
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