If a, b, c are three 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

Question

If a, b, c are three 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: 3: 5 (C) 2: 3: 4 (D) 1: 2 :4.

Tardigrade Admin · Accepted Answer

Since a, b, c are in A.P. ∴ b - a = c - b Also b - a, c - b, a are in G.P. ∴ (c-b)2 = (b-a)a ⇒ (b - a)2 = (b - a)a ⇒ b - a = a ⇒ b = 2a Also c = 2b - a = 2(2a) - a = 3a ∴ a: b: c = a: 2 a: 3a = 1: 2: 3

Q. If a, b, c are three 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

1 : 2 : 3

1 : 3 : 5

2 : 3 : 4

1 : 2 :4.

Since $a, b, c$ are in $A . P$ .
$∴ b - a = c - b$
Also $b - a, c - b, a$ are in $G . P$ .
$∴ (c - b) 2 = (b - a) a$
$\Rightarrow (b - a)^{2} = (b - a) a$
$\Rightarrow b - a = a$
$\Rightarrow b = 2 a$
Also $c = 2 b - a = 2 (2 a) - a = 3 a$
$∴ a : b : c = a : 2 a : 3 a = 1 : 2 : 3$

Q. If a, b, c are three 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

1 : 2 : 3

1 : 3 : 5

2 : 3 : 4

1 : 2 :4.

Since a,b,c are in A.P. ∴b−a=c−b Also b−a,c−b,a are in G.P. ∴(c−b)2=(b−a)a ⇒(b−a)2=(b−a)a ⇒b−a=a ⇒b=2a Also c=2b−a=2(2a)−a=3a ∴a:b:c=a:2a:3a=1:2:3

Since $a, b, c$ are in $A . P$ .
$∴ b - a = c - b$
Also $b - a, c - b, a$ are in $G . P$ .
$∴ (c - b) 2 = (b - a) a$
$\Rightarrow (b - a)^{2} = (b - a) a$
$\Rightarrow b - a = a$
$\Rightarrow b = 2 a$
Also $c = 2 b - a = 2 (2 a) - a = 3 a$
$∴ a : b : c = a : 2 a : 3 a = 1 : 2 : 3$