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 x2, b2, y2 are in AP.

Answer :

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.


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