.Find the sum  of series of even squares in natural series .

S²_e = 2² + 4² + 6² + 8² ……… (2n)²

        = 2² [ 1² + 2² + 3² ………. n² )   where n = N_H/2

     = 4 [ n(n+1)(2n+1)/6] = (2/3)n(n+1)(2n+1)

 Find th sum of series of odd squares in natural series

  S²_o = 1² + 3² + 5² + 7² ……… (2n – 1)²

The sum can be determined by finding out the mean common difference.

With two terms 1² + 3² = 10 =(2+d)   or  d = 8

With three terms  1² + 3² + 5² = 35 = 3 +3d or d = 10 + 2/3  = 8 + (2 +2/3)

With four terms 1² + 3² + 5² + 7² = 84 = 4 +6d or d = 13 +1/3 = 8 + 2 ( 2 + 2/3)

With n terms 1² + 3² + 5² + 7² ……… (2n – 1)²   , n + n(n-1)<d>/2  where the mean <d> = 8 +(n-2)[2 +2/3] = 8 + 8n/3 – 16/3 = (8/3) (n+1)

S²_o = n [ 1 + (n-1) <d>/2]

     = n [ 1 + (n-1)/2 [8/3 (n+1)]

     = n [ 1 +(4/3)(n² -1)] = (n/3) [4n² – 1]