Question

If a, b, c are unequal numbers such that a, b, c are in A.P. and b – a, c – b, a are in G.P, then a : b : c is

(a) 1 : 2 : 3 

(b) 1 : 2 : 4 

(c) 1 : 3 : 4 

(d) 2 : 3 : 4

Answer 1

(a) 1 : 2 : 3

a, b, c are in A.P. ⇒ b – a = c – b               ...(i) 

(b – a), (c – b), a are in G.P ⇒ (c – b)² = (b – a) a        ...(ii)

From (i) and (ii)

⇒ (b – a)² = (b – a)a 

⇒ (b – a) [(b – a) – a] = 0 

⇒ b – a = 0 or b – 2a = 0 

⇒ b = 2a                   (∵ a and b are distinct and b – a ≠ 0)

a, b, c are in A.P. ⇒ 2b = a + c 

⇒ 4a = a + c ⇒ c = 3a. 

∴ a : b : c = a : 2a : 3a = 1 : 2 : 3.