Let S be the set of all values of a 1 for which the mean deviation about the mean of 100 consecutive positive integers a 1 , a 2 , a 3 , … . , a 100 is 25 . Then S is

Step-by-step solution

Given, a 1 be any natural number So, a 1 , a 1 + 1 , a 1 + 2 , … . , a 1 + 99 are values of a i or observation. Now mean of the observation is given by, x ¯ = a 1 + a 1 + 1 + a 1 + 2 + … . + a 1 + 99 100 ⇒ x ¯ = 100 a 1 + ( 1 + 2 + … . . + 99 ) 100 ⇒ x ¯ = a 1 + 99 × 100 2 × 100 ⇒ x ¯ = a 1 + 99 2 Mean deviation about mean = ∑ i = 1 100 x i - x ¯ 100 = 2 99 2 + 97 2 + 95 2 + … . + 1 2 100 = 1 + 3 + … . + 99 100 = 50 2 [ 1 + 99