**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.

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