Question

Let a₁ = 0 and a₁, a₂, a₃, …, an be real numbers such that |a_i| = |a_i − 1 + 1| for all i = 0, 1, 2, …, n. If the AM of the numbers a₁, a₂, a₃, …, an has the value x, then

(A)  x < 1

(B)  x <  1/2

(C)  x ≥ - 1/2

(D)  x ≥ 1

Answer 1

Correct option  (C)  x ≥ - 1/2

|a_i| = |a_{i - 1} + 1| for i = 2,............,n

Squaring we have a²_i = a²_{i - 1} + 2a_{i - 1} + 1 ⇒ a²_i - a²_{i - 1} = 2a_{i - 1} + 1