Question:
If $a, b, c$ are in GP, prove that $a^{2}, b^{2}, c^{2}$ are in GP.
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.
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