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Stuck on a simple Series

  1. Jan 13, 2006 #1
    It's a nice li'l series. The question is to find an expression for the series


    S=1^3-2^3+3^3-4^3+.......-(2m-2)^3+(2m-1)^3

    where m is a subset of N

    I added and subtracted sum of even squares to get

    S={1^3+2^3+3^3+4^3+.....+(2m-1)^3}-{2(2^3+4^3+6^3+...+(2m-2)^3)}

    Replaced the first part by [(2m-1)(2m)/2]^2 to get

    S=[(2m-1)(2m)/2]^2-2(2^3+4^3+6^3+...+(2m-2)^3)

    now All I need to complete the problem is to get the general formula for the sum of cubes of the first n even natural numbers, plug the values and evaluate. Can anyone please help?

    Also, Please let me know if there is an alternative way to solve this problem.
     
  2. jcsd
  3. Jan 13, 2006 #2

    Tx

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    I'd use the sum of an AP to solve this. Unless I've read the question wrong.
     
  4. Jan 13, 2006 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    No, sum of cubes is not an arithmetic progression.

    23+ 43+ 63+ . . .
    = 23+ (2*2)3+ (2*3)3+ ...
    = 23(1+ 23+ 33+...)

    Do you know the formula for the sum of cubes?
     
  5. Jan 13, 2006 #4
    Oh! Thank you. I'm kicking myself for not figuring that out :)
     
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