The number of values of x such that x , x and x are in arithmetic progression is equal to (where, ⋅ denotes the greatest integer function and ⋅ denotes the fractional part function)

Step-by-step solution

We have, 2 x = x + x = x + 2 x ⇒ x = x 2 . Now, 0 ≤ x < 1 ⇒ 0 ≤ x 2 < 1 ⇒ 0 ≤ x < 2 ⇒ x = 0 , 1 for x = 0 ,   x = 0 ⇒ x = 0 for x = 1 ,   x = 1 2 ⇒ x = 3 2 Thus, x = 0 and 3 2 .