Q. 205.0( 3 Votes )

# If a, b, c are in AP, and a, x, b and b, y, c are in GP then show that x^{2}, b^{2}, y^{2} are in AP.

Answer :

To prove: x^{2}, b^{2}, y^{2} 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

⇒ x^{2} = ab … (ii)

As, b,y,c are in GP

⇒ y^{2}= bc … (iii)

Considering x^{2}, b^{2}, y^{2}

x^{2} + y^{2} = ab + bc [From eqn. (ii) and (iii)]

= b (a + c)

= b(2b) [From eqn. (i)]

x^{2} + y^{2} = 2b^{2}

From the above equation we can say that x^{2}, b^{2}, y^{2} are in AP.

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