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    • Thread Starter

    Given that 4r^3 - r^2 (r+1)^2 - (r-1)^2r^2, find E r^3


    r=1; 4 - 0
    r=2; 36 - 144
    r=3; 144 - 36
    r=n; n^2(n+1)^2 - (n-1)^2 n^2

    i ended up with 4Er^3 = 4
    Er^3 = 1

    which is wrong, lol. anyone know where i went wrong?
    • PS Helper

    PS Helper
    I take it by "4r^3 - r^2 (r+1)^2 - (r-1)^2r^2" you meant "4r^3 = r^2 (r+1)^2 - (r-1)^2r^2"?

    Let f(r) = \dfrac{(r-1)^2 r^2}{4}. Then you have that r^3 = f(r+1) - f(r), and so \displaystyle \sum_{r=1}^n r^3 = \sum_{r=1}^n [ f(r+1) - f(r) ]. You can work out which terms stay and which terms disappear using something similar to what you did (but not the same as what you did, because what you did was ultimately wrong :p:). It's easiest to do this in terms of f first, and then plug in the values at the end.

    By 'what you did was ultimately wrong' I mean you seem to have cancelled everything, when you should have a term left at each end. Also, for r=2 and r=3 you've got the same thing, just a factor of -1 apart; so you calculated one of them wrong.
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