Please solve with steps and give a neat solution as soon as possible please

x] can be written as X+{x}    --> {x} is fractional part .. between 0 and 1

therefore , 

lim n→∞  [x]+[2x]+[3x]+.........+[nx]

                                  n2 

is = 

lim n→∞  x+2x+3x+.........+nx + {x}+{2x} ...+{nx}

                                  n2 

= x(1+2+3+..+n)   +   {x}+...{nx}

=x(n(n+1))/2n^2 +  ({x}+...{nx})/n^2

Since {} is only b/w 0 and 1  , the second operand becomes 0 as n tends to ∞

on solving the first part , u get (n2 x +nx)/2n2 = x/2 +x/2n

x/2n becomes 0 as x tends to infinity.

therefore the answer is x/2.