If a, b, c are in G.P., prove that the following are also in G.P.:
(i) a2, b2, c2
(ii) a3, b3, c3
(iii) a2 + b2, ab + bc, b2 + c2
a, b and c are in G.P.
$\therefore b^{2}=a c \quad \ldots \ldots(1)$
(i) $\left(b^{2}\right)^{2}=(a c)^{2} \quad[$ Using $(1)]$
$\Rightarrow\left(b^{2}\right)^{2}=a^{2} c^{2}$
Therefore, $a^{2}, b^{2}$ and $c^{2}$ are also in $\mathrm{G} . \mathrm{P} .$
(ii) $\left(b^{3}\right)^{2}=\left(b^{2}\right)^{3}=(a c)^{3} \quad[$ Using $(1)]$
$\Rightarrow\left(b^{3}\right)^{2}=a^{3} c^{3}$
Therefore, $a^{3}, b^{3}$ and $c^{3}$ are also in G.P.
$(\mathrm{iii})(a b+b c)^{2}=(a b)^{2}+2 a b^{2} c+(b c)^{2}$
$\Rightarrow(a b+b c)^{2}=(a b)^{2}+a b^{2} c+a b^{2} c+(b c)^{2}$
$\Rightarrow(a b+b c)^{2}=a^{2} b^{2}+a c(a c)+b^{2}\left(b^{2}\right)+b^{2} c^{2}$ [Using (1)]
$\Rightarrow(a b+b c)^{2}=a^{2}\left(b^{2}+c^{2}\right)+b^{2}\left(b^{2}+c^{2}\right)$
$\Rightarrow(a b+b c)^{2}=\left(b^{2}+c^{2}\right)\left(a^{2}+b^{2}\right)$
Therefore, $\left(a^{2}+b^{2}\right),\left(b^{2}+c^{2}\right)$ and $(a b+b c)$ are also in G. P.
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