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Summations Type Problem

  1. Dec 30, 2011 #1
    1. The problem statement, all variables and given/known data

    Write down the numbers 1,2,3, ….
    Delete every third number, beginning
    with the third. Write down the
    cumulative sums of the numbers which
    remain. That is:
    1 2 3 4 5 6 7 …
    1 2 4 5 7 …
    1 3 7 12 19 …
    Now delete every second number,
    starting with the second, and write
    down the cumulative sums of what
    remains


    I know that it always ends up as the cubic numbers ie:

    1 8 27 64 etc

    But how would I make a proof of this?


    2. Relevant equations

    Summations of r, r^2 and 1 between 1 and n



    Literally don'tknow how to do it at all!
     
  2. jcsd
  3. Dec 30, 2011 #2

    Dick

    User Avatar
    Science Advisor
    Homework Helper

    Try working forwards and backwards from the sequence just before you did the last cumulative sum yielding the cubes. That was 1,7,19,37,61,... Look at difference between successive elements. That gives you 6,12,18,24,... there's a pretty obvious pattern there. Can you work forward to show the successive sums of that are the cubes? Now can you look back and see how to prove how those differences come about?
     
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