Subject: Series Derivative
Name: Shamus O'Toole
The simplest method is mathematical induction. Let P(n) be the the conjectured identity. Check its validity for n=1. Assume that you have checked it out all the way for n = 1, 2, ..., k-1. Now use that assumption to show the validity for n = k.
That is assume that 1+4+...+(k-1)2 = (k-1)k(2k-1)/6 and then calculate 1+4+..+(k-1)2+k2 = (k-1)k(2k-1)/6 + k2 and verify that it is (k(k+1)(2k+1))/6.Cheers,