In this article in http://www.voofie.com/concept/Mathematics/" [Broken]:
http://www.voofie.com/content/133/repeated-sum-and-partial-difference-equation/" [Broken]
I first gave a standard method in solving the problem (probably the same approach as your theorem). Then I solve it using a...
Use
(k+1)^3 - k^3 = 3k^2+3k+1
and do summation on both side. By telescoping method, only 2 terms left in the left hand side. You need to do a little manipulation on the right hand side though.