Question

If a, b, c are in AP, and a, x, b and b, y, c are in GP then show that x² , b² , y²are in AP.

Answer 1

To prove: x² , b² , y² are in AP. 

Given: a, b, c are in AP, and a, x, b and b, y, c are in GP 

Proof: As, a,b,c are in AP 

⇒ 2b = a + c … (i) 

As, a,x,b are in GP 

⇒ x² = ab … (ii) 

As, b,y,c are in GP 

⇒ y²= bc … (iii) 

Considering x² , b² , y² x ² + y² = ab + bc [From eqn. (ii) and (iii)] 

= b (a + c) = b(2b) [From eqn. (i)] 

x ² + y² = 2b² 

From the above equation we can say that x² , b² , y² are in AP.