If a, b, c are in GP, prove that

Solution:

To prove: $a^{2}, b^{2}, c^{2}$ are in GP

Given: $a, b, c$ are in GP

Proof: As a, b, c are in GP

$\Rightarrow \mathrm{b}^{2}=\mathrm{ac} \ldots$ (i)

Considering $b^{2}, c^{2}$

$\frac{c^{2}}{b^{2}}=$ common ratio $=r$

$\Rightarrow \frac{c^{2}}{a c}[$ From eqn. (i)]

$\Rightarrow \frac{c}{a}=r$

Considering $a^{2}, b^{2}$

$\frac{b^{2}}{a^{2}}=$ common ratio $=r$

$\Rightarrow \frac{\mathrm{ac}}{\mathrm{a}^{2}}$ [From eqn. (i)]

$\Rightarrow \frac{c}{a}=r$

We can see that in both the cases we have obtained a common ratio.

Hence $a^{2}, b^{2}, c^{2}$ are in GP.