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Math Help - Finding the Sum

  1. #1
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    Finding the Sum

     \sum ^m _{j=1} 2^j-2^{j-1}
    [2^1-2^{m+1}][1-2]^{-1}-[2^1-2^m][1-2]^{-1}
    2^{m+1}-2-2^m+2
    2^m

    Can someone pin-point where I went wrong?
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  2. #2
    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by quantoembryo View Post
     \sum ^m _{j=1} 2^j-2^{j-1}
    [2^1-2^{m+1}][1-2]^{-1}-[2^1-2^m][1-2]^{-1}
    2^{m+1}-2-2^m-2
    2^m

    Can someone pin-point where I went wrong?
    2^j-2^{j-1}=2^{j-1} (2-1)=2^{j-1}
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  3. #3
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    Okay, but why cannot one merely apply the summation formula from the very beggining, like I had done?
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  4. #4
    MHF Contributor Also sprach Zarathustra's Avatar
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    Quote Originally Posted by quantoembryo View Post
    Okay, but why cannot one merely apply the summation formula from the very beggining, like I had done?
    \sum ^m _{j=1} 2^j-2^{j-1}=\sum ^m _{j=1}2^j+\sum ^m _{j=1}2^{j-1}=...
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  5. #5
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    Whoops, just realized I made a technical error. I see where I went wrong. Thanks!
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  6. #6
    A Plied Mathematician
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    Just a side note: this sum telescopes.
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