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

$a, b$ and $c$ are in A.P.

$\begin{array}{ll}\therefore 2 b=a+c & \ldots \ldots . \text { (i) }\end{array}$

$a, x$ and $b$ are in G.P.

$\therefore x^{2}=a b \quad \ldots \ldots$ (ii)

And, $b, y$ and $c$ are also in G.P.

$\therefore y^{2}=b c$

Now, putting the values of $a$ and $c$ :

$\Rightarrow 2 b=\frac{x^{2}}{b}+\frac{y^{2}}{b}$

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

Therefore, $x^{2}, b^{2}$ and $y^{2}$ are also in $A$. P.