If a, b, c are in AP, and a, x, b and b, y, c are in GP then show that x2, b2, y2 are in AP.
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
x2 + y2 = ab + bc [From eqn. (ii) and (iii)]
= b (a + c)
= b(2b) [From eqn. (i)]
x2 + y2 = 2b2
From the above equation we can say that x2, b2, y2 are in AP.