Question

If a, b, c are in A.P. and a, x, b and b, y, c are in G.P. show that $x^2,b^2,y^2$ are in A.P.

Answer 1

a, b, c are in A.P.

$\Rightarrow b2=a+c$ ............(1)

a, x, b are in GP

$\Rightarrow x^2=ab$ ..............(2)

And b, y, c are in G.P.

$\Rightarrow y^2=bc$ ...............(3)

Now

$2b^2=x^2+y^2$

$=\left(ab\right)+\left(bc\right)$ [Using (2) and (3)]

$2b^2=b(a+c)$

$2b^2=b\left(2b\right)$

$b{2}^2=2b^2$

LHS = RHS

$\Rightarrow 2b^2=x^2+y^2$

$\Rightarrow x^2,b^2,y^2$ are in A.P.