Question:
If $a, b, c$ are in $A P$, and $a, x, b$ and $b, y, c$ are in GP then show that $x^{2}, b^{2}$, $\mathrm{y}^{2}$ are in AP.
Solution:
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
$\Rightarrow 2 b=a+c \ldots$ (i)
As, a,x,b are in GP
$\Rightarrow x^{2}=a b \ldots$ (ii)
As, b,y,c are in GP
$\Rightarrow \mathrm{y}^{2}=\mathrm{bc} \ldots$ (iii)
Considering $\mathrm{x}^{2}, \mathrm{~b}^{2}, \mathrm{y}^{2}$
$\mathrm{x}^{2}+\mathrm{y}^{2}=\mathrm{ab}+\mathrm{bc}$ [From eqn. (ii) and (iii)]
$=b(a+c)$
= b(2b) [From eqn. (i)]
$x^{2}+y^{2}=2 b^{2}$
From the above equation we can say that $x^{2}, b^{2}, y^{2}$ are in AP.