A number n is randomly selected from the set 1,2,3, ., 1000 . The probability that _ i=1 ^n i^2 _ i=1 ^n i is an integer is

Step-by-step solution

_ i=1 ^n i^2 _ i=1 ^n i = n(n+1)(2 n+1) 6 n(n+1) 2 = 2 n+1 3 For n=1,2,3, . ., 1000 Value of 2 n+1 3 = 3 3 , 5 3 , 7 3 , , 2001 3 respectively. Out of 3 3 , 5 3 , 7 3 , , 2001 3 only first term ( 3 3 =1 ) , fourth term ( 9 3 =3 ), 667^ th term ( 2001 3 =667 ) are integers. Hence, out of 1000 values of 2 n+1 3 , total number of integral values of 2 n+1 3 =333+1=334 Required probability = 334 1000 =0.334