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Simple sum problem

  1. Mar 7, 2013 #1
    (1/1-1/6) + (1/2-1/7) + (1/3-1/8)...+(1/95-1/100)
    This was on the cover of a local paper with the caption "You can't solve this but he can!" The 'he' in the caption was a 12 year old boy. On the next page they gave as his solution 2.2. Two things are clear,
    1.-the boy had the right answer and
    2.-the editor cut the answer to one decimal point for brevity.
    I was wondering, while I drove home from where I saw the cover and second page, what is the actual point one should stop adding the bracketed values to get closest to the abbreviated answer in the paper of 2.2. It took me about a minute to solve it in my head while driving.
    No pen, paper and DEFINITELY no calculator required. Be as brief as possible in your explanation of the simplest solution.
    mathal
     
    Last edited: Mar 8, 2013
  2. jcsd
  3. Mar 8, 2013 #2

    CompuChip

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    My initial thoughts:

    My first observation is that the answer will be 1/1 + 1/2 + 1/3 + 1/4 + 1/5 - 1/96 - 1/97 - 1/98 - 1/99 - 1/100 as all the other values cancel out.

    Using the first 5 terms rounded to 1 or 2 decimals you would get 1 + 0.5 + 0.33 + 0.25 + 0.2 = 2.28.

    If you want to make it more precise you could subtract 5 * 1/100 = 0.05 to get 2.23.

    I checked this using a calculator and it is actually correct to the given precision.

    But I'll be happy to hear a smarter approach.
     
  4. Mar 9, 2013 #3
    This is a reply to the last post. Ignore it if you want to solve this without any help.

    mathal
     
    Last edited: Mar 9, 2013
  5. Mar 9, 2013 #4
    This post contains the answer inside the spoiler.
    For a DOH!! moment read it. I'd stick with the AHA moment, but to each his/her own.
    1+1/2+1/3+1/4+1/5 -2.2 is the sum of the last 5 negative terms needed.
    2+17/60 -(2 +12/60) =5/60
    The last term to add to get closest to the abreviated answer of 2.2 is (1/57-1/62)

    mathal
     
  6. Mar 10, 2013 #5
    I'm going to leave this last message for clarification. CompuChip submitted a message in which he explained the math series in the newspaper that inspired this puzzle. His open comments indicate he felt he was submitting an answer to the puzzle. He was mistaken. The answer is in my last post. If you really need help in understanding what is involved in the math series in the newspaper article read his explanation. It is correct. The puzzle was explained in the opening post after mentioning where the idea came from.
    mathal

    Good puzzles are a gestalt.
     
    Last edited: Mar 10, 2013
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