M2 Mathematical Induction

Use binomial theorem ,
(i+1)^3 = i^3 + 3i^2 + 3i + 1
(i+1)^3 - i^3 = 3 i^2 + 3 i + 1 ...(1)
Denote 1+2+...+n by U , 1^2 + 2^2 + ... + n^2 = V .
Our goal is find V .
taking summation on both sides on (1) , sum from 0 to n .
We have
(n+1)^3 - 1 = 3 V + 3 U + n
By given , U = 0.5n(n+1) ,
So ,
3V = (n+1)^3 - 1 - 3(0.5n)(n+1) - n
6V
= 2(n+1)^3 - 2(n+1) - 3n(n+1)
= (n+1) [ 2(n+1)^2 - 2 - 3n ]
= (n+1)( 2n^2 + 4n + 2 - 2 - 3n )
= n(n+1)(2n+1)
Thus , V = n(n+1)(2n+1) / 6