Subject: Series Derivative

Name: Shamus O'Toole
Who is asking: Teacher
Level: Secondary

Question:
How do you derive that for the series 1+4+9+16+25.. that

S(n)=(n(n+1)(2n+1))/6

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.