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How much is this sum?

  1. Jan 29, 2013 #1
    exp.png

    the answer is 3^2048. How do I get there?
     
  2. jcsd
  3. Jan 29, 2013 #2

    AlephZero

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    Since you are given the answer, use that information!

    You have to prove that
    ##(2+3)(2^2+3^2)\cdots(2^{2048} + 3^{2048}) + 2^{4096} - 3^{4096} = 0##
    Now, think what you can do with ##2^{4096} - 3^{4096}## ...
     
  4. Jan 30, 2013 #3
    i don't know, what can i do :S?
     
  5. Jan 30, 2013 #4
    and the answer is not given, it's multiple choice
     
  6. Jan 30, 2013 #5

    mathman

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    a2 - b2 = (a-b)(a+b)

    Start with a = 22048 and b = 32048
    next repeat with a = 21024 and b = 31024
    etc.
    At the end you will have (2-3)(2+3). Just be careful with the sign.
     
    Last edited: Jan 30, 2013
  7. Jan 30, 2013 #6
    but i got a plus sign not a minus sign...
     
  8. Jan 31, 2013 #7

    HallsofIvy

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    No, its a minus sign:
    ##(2+3)(2^2+3^2)\cdots(2^{2048} + 3^{2048}) + 2^{4096} - 3^{4096} = 0##
    AlephZero was referring to the last pair on the left.
     
  9. Jan 31, 2013 #8
    now i get it. thanks!
     
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