Number Theory

In sequences of this type, it is best to find, explicitly, the first few terms of the sequence and attempt to find a "closed form" of the sequence, which will make it much easier to sum it up.

However, it is not always easy to find such "closed forms." But, there are computer algorithms that aid greatly in finding such algorithms. I happened to have written one such algorithm in C++ and is very good at finding patterns.

(2, 40, 168, 440, 910, 1632, 2660, 4048, 5850)
Including the above terms in the program, my computer was able to come up with this very convenient "closed form" of the sequence:  (9n^3 - 9n^2 + 2n).

 Using the above closed form, we can easily sum up the sequence as follows:
Sumfor(n, 1, 999, (9n^3 - 9n^2 + 2n) = 2,242,507,747,500 - which is the sum of this sequence.

[1*2*3]/3 + [4*5*6]/3 + [7*8*9]/3 + [ 10*11*12] / 3 +{13*14*15]/3 +..........+ [997*998*999]/3

1*2*1   + 4*5*2   + 7*8*3   + 10*11*4  + 13*14*5 +  997*998*333  =

1*2  + 2*4*5     +  3*7*8    + 4*10*11  +   5*13*14    +.........+  333*997*998   =

      331

2  + ∑    [ (n + 2)* (3n + 4)(3n + 5) ]   =  27,722,083,500

      n = 0

Sorry CPhill !!!. I simply used all 999 terms instead of just 1/3 of them !!.

Sumfor(n, 1, 333, (9n^3 - 9n^2 + 2n) = 27,722,083,500 - which is the sum of this sequence.