<PRE><B>Question 12205</B>
 To find the sum S =
 1 x 2 x 3 - 2^3 + 3 x 4 x 5 - 4^3 + ... + 9999 x 10000 x 10001 x - 10000^3
 
 Consider the general form of every two consecutive terms as:
 1 * 2 * 3 - 2^3 = 2(1*3 - 2*2) = -2,
 3 * 4 * 5 - 4^3 = 4(3*5 - 4^2) = -4,
 ...
 (2n-1)2n(2n+1) - (2n)^3= 2n[(2n-1)(2n+1) - 4n^2) = 2n(4n^2-1- 4n^2) = -2n,
 ....
 9999 * 10000 * 10001  - 10000^3 = - 10000 (last two terms)

 Totally, there are 10000/2 = 5000 such pairs.

 Hence, S = -(2+4+6+...+10000) = -2(1+2+3+...+5000)
 = -2(5001*5000/2) [why ?]
 = -25005000

 Not really hard.

 Kenny 
 </PRE>