If [x] denotes the greatest integer x , then _ n - 1 n^3 [1^2 x ]+ [2^2 x ]+ [3^2 x ]+ + [n^2 x ] =

Step-by-step solution

aligned & _ n 1 n^3 [1^2 x ]+ [2^2 x ]+ [3^2 x ]+ + [n^2 x ] & = _ n 1 n^3 _ r=1 ^n [r^2 x ]= _ n 1 n^3 _ r=1 ^n (r^2 x- r^2 x ) & = _ n 1 n^3 (x _ r=1 ^n r^2- _ r=1 ^n r^2 n ) & = _ n x n(n+1)(2 n+1) n^3 6 - _ r=1 ^n r^2 n n^3 & = _ n ( x 6 n n n+1 n (2 n+1) n )- _ n _ r=1 ^n r^2 n n^3 & = x 6 1 2-0= x 3 aligned