if a,b,c are in G.P. prove that $a^{2} + b^{2}, a b + b c, b^{2} + c^{2}$ are also in G.P.

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Solution

Given,

$a, b, c$ are in G.P.

$r$ is the common ratio

$b = a r, c = a r^{2}$

if,

$\frac{a b + b c}{a^{2} + b^{2}} = \frac{b^{2} + c^{2}}{a b + b c}$

$\Rightarrow \frac{a (a r) + (a r) (a r^{2})}{a^{2} + (a r)^{2}} = \frac{(a r)^{2} + (a r^{2})^{2}}{a (a r) + (a r) (a r^{2})}$

$\frac{a^{2} r (1 + r^{2})}{a^{2} (1 + r^{2})} = \frac{a^{2} r^{2} (1 + r^{2})}{a^{2} r (1 + r^{2})}$

$\Rightarrow a^{2} + b^{2}, a b + b c, b^{2} + c^{2}$ are in GP.

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